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Billiard Ball Dynamics A.C. Baroudy acbarou@inco.com.lb Introduction The dynamics involved in the motion of the billiard ball is part of the rotation dynamics. This is where angular momentum L, angular acceleration \316\261, and torque \316\223 are used. Note that the torque is the equivalent of the force F = ma so it is expressed as \316\223 =I\316\261 where I is the mass moment of inertia (equivalent to the mass m in F = ma) about a given axis of rotation and \316\261 the angular acceleration equivalent to the linear acceleration a. The ball spin and forward translation depend on the point of impact of the cue on the ball as related to the center of mass(CM): Considering only shots where the cue hits the ball always in its vertical median plane and in a horizontal direction we state a few facts about the motion of the ball in this setting, there explanation will follw the dynamics of the ball: 1- If the cue hits the ball in the direction of the center of the ball this latter will go on sliding and rolling for a certain time then rolling alone. 2- If the cue hits the ball well above the ball center then the ball will have say, a CW spin and a forward sliding motion. 3- If the cue hits the ball well below the ball center then the ball will have a CCW spin and a forward sliding motion. 4- If R is the radius of the bal and the cue hits the ball at 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 R above the ball center, then the ball will have say, a CW spin causing a forward rolling with no sliding. Here we have 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 at any time i.e.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1GIzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUY0NiVRInhGJ0Y3RjovRjtRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQi1GIzYkLUkjbW9HRiQ2LVEtJlJpZ2h0QXJyb3c7RidGQi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTy8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0ZPLyUobGFyZ2VvcEdGTy8lLm1vdmFibGVsaW1pdHNHRk8vJSdhY2NlbnRHRk8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZobkZCLyUnYWNjZW50R0Y5RkI=\342\206\222(along +x) & 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 \342\206\220(along -x) We will see why later on. 5- If the cue hits the ball at 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 R below the ball center then the ball will have say, a CCW spin and a forward rolling with sliding. Here we have 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 at any time. Both being along +x Our problem is to analyse the motion of a cue ball where: 1- M is the ball's mass and its mass moment of inertia = I = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYkUSI1RidGMkYyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Y/LyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEjTVJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JEYuRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI= around an axis passing through its center of mass. 2- 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 is the initial angular velocity. The shot being well above the critical point 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R from the center of the ball. 3- LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYvNiVRImtGJ0YyRjhGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRidGOA== being the coefficient of kinetic friction. 4- LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== forward initial velocity of CM. As stated above we consider only shots where the cue hits the ball always in its vertical median plane and in a horizontal direction. To be more specific the cue has just hit the ball well above the 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 critical distance above the center of mass. In this case at P, point of contact of the ball with the table, we must have : 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 i.e. the translation velocity is less than the velocity of rotation. See equation (8') below. As for the direction of LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1JI21uR0YkNiRRIjBGJy9GM1Enbm9ybWFsRidGPC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn it is the same as +x while that of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIjBGJ0Y1RjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYlUSJSRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGNQ== it is along -x in this setting. Our coordinates are the principal axes with CM being the origin O. This right hand triade is considered as fixed in space: ** the x-axis along the forward translation, ** the y-axis to the left of x-axis, ** the z-axis directed upward. The ball, as we mentioned above, will have a spin and a sliding motion at the very start. The friction force \342\204\261, according to the law of kinetic, acts in a direction opposite to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYnUSJjRidGMkY1RjdGOkYyL0Y4RjxGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y4USdub3JtYWxGJw== the velocity of P. This force will operate until LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYlUSJjRidGNS9GOFEnaXRhbGljRidGMi9GOEY8RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GOFEnbm9ybWFsRic= = 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 = 0 then it vanishes, where 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 = translation velocity of CM & P, & 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 = rotation velocity at P. We propose to : 1- find the forward accelration a and the angular acceleration \316\261 before the pure rolling occurs, 2- find the distance d the ball has to travel till the friction force \342\204\261 vanishes i.e. the moment when the sliding ceases and the ball has a pure rolling motion where v = R\317\211. 3- find the work done by the friction force of sliding \342\204\261 on the ball. 4- find the magic number 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. Finding the equations of motion The forces acting on the ball at P are: a- the weight = -Mg along -z, b- the normal force of contact at P = \356\224\221 along +z, c- the horizontal force of contact at P = force of friction = \342\204\261, it is always acting to oppose point P velocity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYnUSJjRidGMkY1RjdGOkYyL0Y4RjxGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y4USdub3JtYWxGJw== hence it is along +x in the present setting. Then we have : 1- along z-axis: -Mg+\356\224\221 = 0, since the ball CM has no vertical acceleration. This gives \356\224\221 = Mg. 2- along x-axis: \342\204\261= Ma = 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. (1) Now the kinetic coefficient of friction is defined as the ratio of the two forces of contact involved in the friction: the horizontal \342\204\261 and the normal \356\224\221 i.e. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJrRidGMkY4RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUkjbW9HRiQ2LVEiPUYnRjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZaLUZHNi1RIn5GJ0Y4RkpGTEZORlBGUkZURlYvRllRJjAuMGVtRicvRmZuRltvLUkmbWZyYWNHRiQ2KC1GLzYlUSgmIzg0OTc7RicvRjNGNy9GOVEnaXRhbGljRictRiM2JC1GLzYlUSkmIzU4NjQxO0YnRmNvRmRvRjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmBwLyUpYmV2ZWxsZWRHRjRGOA== = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRI01hRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtRi82JVEjTWdGJ0YyRjUvRjZRJ25vcm1hbEYnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnRj0= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYlUSJnRidGMkY1L0Y2USdub3JtYWxGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y9, thus we have: a = LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEia0YnRi9GNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==g. (2) The ball can rotate around any axis through its CM. Since \342\204\261 has a moment arm relative to CM = R then a torque is present. In vector notation this torque is 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUSlSJiM4NDk3O0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JJm1vdmVyR0YkNiUtRkc2J1EieUYnLyUlYm9sZEdGTEZKL0YwUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GIzYkLUYsNi1RLSZSaWdodEFycm93O0YnRi9GMkY1L0Y4RkxGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZdb0YvLyUnYWNjZW50R0ZMLUYsNi1RIUYnRi9GMkY1RjdGOUY7Rj1GP0Zcb0Zeb0Yv. Thus it is negative in the direction of -y with magnitude: \316\223 = 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 = R\342\204\261 = RMLUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEia0YnRi9GNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==g =I\316\261 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYkUSI1RidGMkYyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Y/LyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEjTVJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JEYuRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=\316\261 . and \316\261 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRJyYjOTY5O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1JI21vR0YkNi1RLCZDZW50ZXJEb3Q7RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRk5GNS8lJ2FjY2VudEdRJXRydWVGJ0Y1 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiNUYnRi8tRiM2JC1GSjYkUSIyRidGL0YvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZXLyUpYmV2ZWxsZWRHRjRGLw==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 (3) The two equations: I- \342\204\261 = M LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEia0YnRi9GNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==gLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNidRInhGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjQvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUYjNiQtSSNtb0dGJDYtUS0mUmlnaHRBcnJvdztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkcvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGRy8lKGxhcmdlb3BHRkcvJS5tb3ZhYmxlbGltaXRzR0ZHLyUnYWNjZW50R0ZHLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkZDLyUnYWNjZW50R0Y0RkM=, II- \316\223=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 define the motion of the ball. It is interesting to note how the angular velocity \317\211 and the angular acceleration LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNidRJyYjOTY5O0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JC1JI21vR0YkNi1RLCZDZW50ZXJEb3Q7RicvRjlRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGVUZDLyUnYWNjZW50R0Y0RkM= are of opposite directions. The torque inherits the direction of the angular acceleration! From the above two equations we get the acceleration of contact point P: Starting with 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 we get : 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 = LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEia0YnRi9GNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==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g LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21pR0YkNidRInhGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjEvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUklbXJvd0dGJDYkLUkjbW9HRiQ2LVEtJlJpZ2h0QXJyb3c7RicvRjVRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlRGQS8lJ2FjY2VudEdGMQ==+ LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiNUYnRi8tRiM2JC1GSjYkUSIyRidGL0YvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZXLyUpYmV2ZWxsZWRHRjRGLw==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= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI3RidGL0YvLUYjNiQtRkw2JFEiMkYnRi9GLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWS8lKWJldmVsbGVkR0Y0Ri8=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, (3.0) hence the acceleration of point P has the same direction as that of \342\204\261 which, as we know, is along +x & opposite to that of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==. Thus at time t, LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn is negative along -x : 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI3RidGL0YvLUYjNiQtRkw2JFEiMkYnRi9GLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWS8lKWJldmVsbGVkR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJrRidGMkY4RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYvNiVRI2d0RicvRjNGNy9GOVEnaXRhbGljRidGOA== where 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 since in the present setting, as we stated above, we have 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 & LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GIzYlLUYsNiVRImNGJ0YvRjItRjY2JC1GLDYlUSJvRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkJGRA== is negative . Hence: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI3RidGL0YvLUYjNiQtRkw2JFEiMkYnRi9GLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWS8lKWJldmVsbGVkR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJrRidGMkY4RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYvNiVRI2d0RicvRjNGNy9GOVEnaXRhbGljRidGOA== (3.1) If we call LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInJGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSFGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRldGPQ== the rotation velocity & LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI3RyRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= the translation velocity then: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI3RyRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUZDNi1RIn5GJ0Y9RkZGSUZLRk1GT0ZRRlMvRlZRJjAuMGVtRicvRllGaG4tRiw2JUYuLUYjNiQtSSNtbkdGJDYkUSIwRidGPUY9Rj8tRkM2LVEiK0YnRj1GRkZJRktGTUZPRlFGUy9GVlEsMC4yMjIyMjIyZW1GJy9GWUZmby1GLzYlUSNhdEYnRjJGNUZaRlpGQkZqbkZib0Y9LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEia0YnRi9GNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==g t \342\206\222 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI3RyRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUYsNi1RIn5GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRJjAuMGVtRicvRkVGSi1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRInZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjBGJ0YvRi8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLDYtUSIrRidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRl9vRi8=LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEia0YnRi9GNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==g t (3.11) and from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdWJHRiQ2JUYrLUYjNiQtSSNtbkdGJDYkUSIwRidGMkYyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiK0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHRjRGL0Yvt we get \317\211R = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnRjVGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi9RKCZtaW51cztGJy8lJWJvbGRHRjEvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjwvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGUC1GNjYtUSFGJy9GM1Enbm9ybWFsRidGP0ZCRkRGRkZIRkpGTC9GT1EsMC4yNzc3Nzc4ZW1GJy9GUkZZRlY=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. (3.12) Hence (3.1) becomes:: 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, (3.2) i.e. 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 (3.3) Note that (3.2) is valid as long as t is still less than the time when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUkjbW5HRiQ2JFEiMEYnRj1GPQ==. This time = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRJmZpbmFsRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEhRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXRj0= is obtained from (3.1) by setting LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn equal to 0. It is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRJmZpbmFsRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIyRidGPS1GIzYkLUZobjYkUSI3RidGPUY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zlby8lKWJldmVsbGVkR0ZILUZlbjYoLUYjNictRiw2JS1GLzYlUScmIzk2OTtGJy9GM0ZIRj0tRiM2JC1GaG42JFEiMEYnRj1GPUY/LUYvNiVRIlJGJ0YyRjUtRkM2LVEoJm1pbnVzO0YnRj1GRkZJRktGTUZPRlFGUy9GVlEsMC4yMjIyMjIyZW1GJy9GWUZgcS1GLDYlLUYvNiVRInZGJ0YyRjVGZHBGP0Y9LUYjNiUtRiw2Ji1GLzYmUSUmbXU7RidGY3AvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjRGPS1GIzYkLUYvNiVRImtGJ0ZjcEY9Rj1GPy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYvNiVRImdGJ0YyRjVGPUZgb0Zjb0Zmb0Zob0Y9. (3.4) From that point on we must have: 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 . Finding the distance d II- To find the distance d we start with equation (3.4) which gives the time when \342\204\261 vanishes. The time is the upper limit of the integration of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUTAmRGlmZmVyZW50aWFsRDtGJ0Y5L0Y8USZ1bnNldEYnL0Y/RlMvRkFGUy9GQ0ZTL0ZFRlMvRkdGUy9GSUZTRkpGTS1GLDYlUSJ0RidGL0YyRjk= : d = 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 = 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g 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==. Substituting the value of t from (3.4) in the above equation we get the required distance: d = 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 + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JJm1mcmFjR0YkNigtRiM2JC1JI21uR0YkNiRRIjFGJ0YvRi8tRiM2JC1GTjYkUSIyRidGL0YvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zlbi8lKWJldmVsbGVkR0Y0Ri9GK0YvLUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEia0YnRi9GNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==g 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 (5) The work done by \342\204\261 on the ball The work done by \342\204\261 on the ball is the difference between the initial energy LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImlGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== and the sum of the rotation energy LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInJGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== & the translation energy LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI3RyRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= of the ball. We need to get \317\211 & v at 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 i.e. when \342\204\261 vanishes and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== = \317\211R. From (3.12) we have \317\211R = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnRjVGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi9RKCZtaW51cztGJy8lJWJvbGRHRjEvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjwvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGUC1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSI1RicvRjNRJ25vcm1hbEYnLUYjNiQtRlc2JFEiMkYnRlpGWi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGQUZaLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JS1JJW1zdWJHRiQ2Ji1JI21pR0YkNiZRJSZtdTtGJy8lJ2l0YWxpY0dGNC8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJ0YvLUYjNiQtRkw2JVEia0YnRk9GL0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GTDYlUSJnRicvRlBGUy9GMFEnaXRhbGljRidGL0YrRi8=t . We substitute the value of t = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRJmZpbmFsRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEhRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXRj0= in it then solve for \317\211 . We get \317\211 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYkUSI3RidGMkYyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Y/LyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkjbW9HRiQ2LVEhRidGMi8lJmZlbmNlR0ZELyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlktRiw2KC1GIzYkLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiotRi82JFEiMkYnRjItRkY2LVEifkYnRjJGSUZLRk1GT0ZRRlNGVS9GWFEmMC4wZW1GJy9GZW5GaW8tSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSlSJm9tZWdhO0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy1GIzYkLUYvNiRRIjBGJ0YyRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GRjYtUSIrRidGMkZJRktGTUZPRlFGU0ZVL0ZYUSwwLjIyMjIyMjJlbUYnL0ZlbkZjcS1GLzYkUSI1RidGMkZlby1GXHA2JS1GX3A2JVEidkYnRmJwRmVwRmdwRlxxRjJGMi1GIzYkRkVGMi8lMXN1cGVyc2NyaXB0c2hpZnRHRl5xRjItRiM2JS1GX3A2JVEiUkYnL0ZjcEZERjJGZW9GMkY6Rj1GQEZCRjI=. Since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== = \317\211R at t = LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSZmaW5hbEYnRi9GMi9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== then we use it instead of getting v from (3.11). 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+ LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYkUSIyRidGMkYyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Y/LyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEjTXZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JC1GLzYkUSIwRidGMkYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlgvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRjI= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI3RyRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGPS1GIzYkLUZobjYkUSIyRidGPUY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zlby8lKWJldmVsbGVkR0ZILUklbXN1cEdGJDYlLUYvNiVRI012RidGMkY1RltvLyUxc3VwZXJzY3JpcHRzaGlmdEdGQUZCRlotRltwNiUtRi82JVEjTVJGJ0YyRjVGW29GYHAtRltwNiUtRi82JVEnJiM5Njk7RicvRjNGSEY9RltvRmBwRkJGWi1GQzYtUSFGJ0Y9RkZGSUZLRk1GT0ZRRlNGVUZYLUYjNiUtRi82JVEiTUYnRjJGNS1GZW42KC1GIzYkLUZbcDYlLUkobWZlbmNlZEdGJDYkLUYjNipGXW8tRkM2LVEifkYnRj1GRkZJRktGTUZPRlFGUy9GVlEmMC4wZW1GJy9GWUZkci1GLDYlLUYvNiVRKVImb21lZ2E7RidGMkY1LUYjNiQtRmhuNiRRIjBGJ0Y9Rj1GPy1GQzYtUSIrRidGPUZGRklGS0ZNRk9GUUZTL0ZWUSwwLjIyMjIyMjJlbUYnL0ZZRmRzLUZobjYkUSI1RidGPUZgci1GLDYlLUYvNiVRInZGJ0YyRjVGW3NGP0Y9Rj1GW29GYHBGPS1GIzYlLUZobjYkUSM0OUYnRj1GYHJGPUZgb0Zjb0Zmb0Zob0Y9Rl1xRj0= Work of \342\204\261 on the ball: 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 = 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 + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYkUSIyRidGMkYyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Y/LyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEjTXZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JC1GLzYkUSIwRidGMkYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlgvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRjI=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 (6) The particular point located at 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 Why hitting the ball with the cue at this level will send the ball into pure rolling with no sliding? To understand this situation we must remember that with pure rolling the point of contact P on the table moves along the rim of the ball the same distance as the center of the ball moves forward in any given time, in other words the point of contact must have the same mgnitude for these two velocities but opposite in direction. Hence we must have in case of rolling without slipping 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 thus the condition for pure rolling is that the velocity of the point of contact LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUkjbW5HRiQ2JFEiMEYnRj1GPQ==. We need to get a relation between the velocity of translation and that of rotation by using the concept of angular momentum, linear momentum & the associated impulses. We consider the ball as being hit by the cue at a height h above the point of contact P. Our coordinates are x-axis along the direction of translation , y-axis parallel to the axis of rotation and z-axis is passing through the center of the ball and upward positive. I- The linear momentum and the impulse of F Here the impulse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkRmR0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMRjk=of the impact force F for a short time \316\224t is equal to the change in linear momentum from MLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== to MLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInRGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== : 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 i.e. \342\206\222 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. Integrating this last equation from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZYRj4=to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== we get : 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 . (6') Since \316\224t is very small of the order of 0.01 second and at t = 0, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSFGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGPg== = 0, then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== is the initial velocity at the end of the impact. We call it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== and we have: 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 . (6") Since the translation is along x-axis and F = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNidRImlGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjQvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUYjNiQtSSNtb0dGJDYtUS0mUmlnaHRBcnJvdztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkcvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGRy8lKGxhcmdlb3BHRkcvJS5tb3ZhYmxlbGltaXRzR0ZHLyUnYWNjZW50R0ZHLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkZDLyUnYWNjZW50R0Y0RkM=then equation (6") is the only scalar equation for the linear momentum at the very start: 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 . (6"') II- The angular momentum and the impulsive torque For the angular momentum of the ball we need the arm moment of the force delivered by the cue. It is ( h - R ) where h is the height above contact point P. The change \316\224L in the angular momentum (L) : \316\224L = angular impulse = \316\223\316\224t \316\224L = product of the torque and the interval of time during which the torque is acting. III- Finding the torque \316\223 and its impulse Note that F the impulsive force is along +x-axis and ( h - R ) as a vector is in the direction of +z ( upward ) so that the cross product LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUYxNiVRIlJGJ0Y0RjdGPkY+LUkmbW92ZXJHRiQ2JS1GMTYnUSJ6RicvJSVib2xkR0Y2RjQvRjhRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUYjNiQtRjs2LVEtJlJpZ2h0QXJyb3c7RidGPkZARkMvRkZGNkZHRklGS0ZNL0ZQUSwwLjI3Nzc3NzhlbUYnL0ZTRmVvRj4vJSdhY2NlbnRHRjYtRjs2LVEoJnRpbWVzO0YnRj5GQEZDRkVGR0ZJRktGTUZPRlItRjE2JVEiRkYnRjRGNy1GIzYkLUZYNiUtRjE2J1EieEYnRmduRjRGaW5GW29GXm9GZ29GPi1GOzYtUSFGJ0Y+RkBGQ0ZFRkdGSUZLRk1GZG9GZm9GPg== is a vector in the direction of +y. This indicates that we take the change in the angular momentum along y-axis. Here, however, \317\211 is a vector already along y-axis & the mass moment of inertia I around y-axis still has the same expression as used above, thus the component of I\317\211 on y-axis equal I\317\211 and the change in angular momentum LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJiM5MTY7TEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInlGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== is: 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 . Since at t = 0, 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= 0 then : 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. (7) Since \316\224t is very small of the order of 0.01 second and at t = 0, LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXJvd0dGJDYkLUYjNiUtSSNtaUdGJDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GLDYkLUYxNiVRInlGJy9GNVEldHJ1ZUYnL0Y4USdpdGFsaWNGJ0Y3LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNy1GLDYkLUkjbW5HRiQ2JFEiMEYnRjdGN0ZD = 0 then 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 is the angular velocity at the end of the impact. We call it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHRjRGLy1GIzYkLUkjbW5HRiQ2JFEiMEYnRi9GLy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRi8= and we have: 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 . (7') To compare these two velocities we multiply both sides of (6"') by the factor (h - R) and equate the LHS of the result to the LHS of (7') : 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 \342\206\222 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, 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 (8) 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 (8') This last equation shows that 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 i.e. 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 when 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 . The velocity of the point of contact P: 1- when h > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiN0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y0LUYjNiQtRjE2JFEiNUYnRjRGNC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGQS8lKWJldmVsbGVkR1EmZmFsc2VGJy1JI21pR0YkNiVRIlJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNVEnaXRhbGljRidGNA==, is 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 and it is, like LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEpJm9tZWdhO1JGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=, negative along -x with 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 , 2- when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JkYrLUYjNiYtSSNtaUdGJDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RIjxGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUZLNiVRImhGJ0ZORlFGL0YrRi9GK0Yv < 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, is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUYsNiVGLi1GIzYkLUYvNiVRInhGJ0YyRjVGPUY/LUZDNi1RKCZtaW51cztGJ0Y9RkZGSUZLRk1GT0ZRRlMvRlZRLDAuMjIyMjIyMmVtRicvRllGX28tRkM2LVEifkYnRj1GRkZJRktGTUZPRlFGUy9GVlEmMC4wZW1GJy9GWUZlby1GLzYlUSgmIzk2OTtSRidGMkY1Rj0= and it is, like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZXRj0=, positive along +x with 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. ** Hence for h \342\251\276 R we have using (4.2) & (4.3): 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 , (9) and at t = 0 i.e. just at the end of the impact we have: 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 (9') ** When h < R we need recalculate the torque and use 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 since for h < R both 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 are positive in the direction of +x. The torque is 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 = 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 hence it is along -y. Thus we need take the change in angular momentum along -y. It just happend that when h < R ,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRJyYjOTY5O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1JI21vR0YkNi1RLSZSaWdodEFycm93O0YnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPRjUvJSdhY2NlbnRHRkRGNQ== is already along -y so that the impulsive torque integral 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 and the change in angular momentum LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSkmRGVsdGE7TEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GIzYkLUZKNiVRInlGJ0ZNRlBGLy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRi8= are related by the same equation (7) except that we use (R - h ) here instead of ( h - R ). Using 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 as we said above we get 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 . This leads to the same formula as (9) & (9') above. Thus equations (9) & (9') for 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 are valid for all cases of h i.e. h \342\251\276 R or h \342\251\275 R giving 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 at t = t and t = 0 respectively. As a check for the correctness of (9') we see that : ** for h = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JJm1mcmFjR0YkNigtRiM2Ji1JI21uR0YkNiRRIjdGJ0YvLUYsNi1RMSZJbnZpc2libGVUaW1lcztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRJjAuMGVtRicvRkVGVS1JI21pR0YkNiVRIlJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGLy1GIzYkLUZONiRRIjVGJ0YvRi8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmRvLyUpYmV2ZWxsZWRHRjRGL0YrRi8=, 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 which is correct since from the start i.e. end of impact, both LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJ4RidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn are equal but opposite in direction thus 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 . ** for h = 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, 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 which is also correct since in this case both LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJ4RidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn are equal and in the same direction thus 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 = 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 ** for h = R, 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 which is correct since there is no torque & LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSFGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGPg== is P's only velocity. As we said above, for pure rolling to set in from the start, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZXRj0=must be equal to zero i.e. 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 ( by setting equation(9') = 0) .Hence the cue must hit the ball at the particular point located LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYkUSI1RidGMkYyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Y/LyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRkQvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGXG9GMg== above CM! Predicting the behavior of the ball from equation (9) & 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 = 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 The behavior of the ball can be predicted from equation (9) and: 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 = 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 (10) as well as the law of kinetic friction as long as there is a slipping between the ball and the table. This law tells us that \342\204\261 acts in a direction opposite the velocity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYnUSJjRidGMkY1RjdGOkYyL0Y4RjxGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y4USdub3JtYWxGJw== of the point of contact P. Note that 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 = 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 equals the algebraic sum of both velocities at P at any time. Eight cases are to be considered. See Figure-1: I- h = 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 where 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: From (9) we have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== = 0 hence \342\204\261 = 0. Motion from the start is a pure rolling. II- High shot when h > 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 where 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: From (9) we have LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn < 0 hence along -x & \342\204\261 along +x. Motion from the start: rolling + sliding. \342\204\261 will increase LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI3RyRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= & decrease LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInJGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== till we have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYlUSJjRidGNS9GOFEnaXRhbGljRidGMi9GOEY8RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GOFEnbm9ybWFsRic= = 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 = 0. At this moment the ball is in pure rolling. III- Low shot when R < h < 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 where 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: From (9) we have LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn > 0 along +x & \342\204\261 along -x. Motion from the start: rolling + sliding. \342\204\261 will increase LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn & decrease LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI3RyRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= till we have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYlUSJjRidGNS9GOFEnaXRhbGljRidGMi9GOEY8RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GOFEnbm9ybWFsRic= = 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 = 0. At this moment the ball is in pure rolling. IV- h = R where 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: From (9) we have LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn = LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn = 0, hence along +x & \342\204\261 along -x. Motion from the start: sliding. \342\204\261 will increase LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn & decrease LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSN0ckYnRi9GMi9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== so the ball will have rolling & sliding till we have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYlUSJjRidGNS9GOFEnaXRhbGljRidGMi9GOEY8RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GOFEnbm9ybWFsRic= = 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 = 0. At this moment the ball is in pure rolling. V- Tangential shot where h = 2R , 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: LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn = LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn & LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSN0ckYnRi9GMi9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== = 0, hence LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn along -x & \342\204\261 along +x. Motion from the start: rolling . \342\204\261 will increase LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSN0ckYnRi9GMi9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== & decrease LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn causing sliding along with the rolling till we have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYlUSJjRidGNS9GOFEnaXRhbGljRidGMi9GOEY8RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GOFEnbm9ybWFsRic= = 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 = 0. At this moment the ball is in pure rolling. 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VI- h = critical point 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 where 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: From (9) we have LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn = 2 LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSN0ckYnRi9GMi9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==& LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn = LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSN0ckYnRi9GMi9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==, hence along +x & \342\204\261 along -x. Motion from the start: rolling CCW & sliding. From (3.1) 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI3RidGL0YvLUYjNiQtRkw2JFEiMkYnRi9GLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWS8lKWJldmVsbGVkR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJrRidGMkY4RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYvNiVRI2d0RicvRjNGNy9GOVEnaXRhbGljRidGOA== we have 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI3RidGL0YvLUYjNiQtRkw2JFEiMkYnRi9GLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWS8lKWJldmVsbGVkR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJrRidGMkY4RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYvNiVRI2d0RicvRjNGNy9GOVEnaXRhbGljRidGOA== since both have the same direction along +x while \342\204\261 is along -x. Which at time \317\204 = 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUkjbW5HRiQ2JFEjMC5GJ0Y9Rj0= Thus the ball keeps rolling and slinding till at time \317\204 it stops. VII- 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 < h < R and 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: From (9) we have LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn >0 , hence along +x & \342\204\261 along -x. Motion from the start: rolling CCW & sliding. from 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI3RidGL0YvLUYjNiQtRkw2JFEiMkYnRi9GLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWS8lKWJldmVsbGVkR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJrRidGMkY4RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYvNiVRI2d0RicvRjNGNy9GOVEnaXRhbGljRidGOA== = 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 \342\204\261 will decrease both LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn & LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSN0ckYnRi9GMi9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== but LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNiQtRiw2JVEickYnRjIvRjVRJ2l0YWxpY0YnL0Y1USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn being smaller than LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNiQtRiw2JVEjdHJGJ0YyL0Y1USdpdGFsaWNGJy9GNVEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== becomes 0 first then it reverses its sign where \342\204\261 is causing the ball to turn CW instead of CCW while still increasing it and further decreasing LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNiQtRiw2JVEjdHJGJ0YyL0Y1USdpdGFsaWNGJy9GNVEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== till we get the condition for pure rolling LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYlUSJjRidGNS9GOFEnaXRhbGljRidGMi9GOEY8RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GOFEnbm9ybWFsRic= = 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 = 0. Here rotation is reversed. VIII- h < 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 and 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: From (9) we have LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJjRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn >0 , hence along +x & \342\204\261 along -x. Motion from the start: rolling CCW & sliding. from 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSI3RidGL0YvLUYjNiQtRkw2JFEiMkYnRi9GLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWS8lKWJldmVsbGVkR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJrRidGMkY4RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYvNiVRI2d0RicvRjNGNy9GOVEnaXRhbGljRidGOA== = 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 \342\204\261 will decrease both LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJyRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn & LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSN0ckYnRi9GMi9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== but LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNiQtRiw2JVEjdHJGJ0YyL0Y1USdpdGFsaWNGJy9GNVEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== being smaller than LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSVtcm93R0YkNiQtRiw2JVEickYnRjIvRjVRJ2l0YWxpY0YnL0Y1USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn becomes 0 first then it reverses its sign. Thus the ball stops advancing and change translation from forward to backward . \342\204\261 will continue increasing this backward translation while it is reducing 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 till we get the condition for pure rolling LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ji1GLzYlUSJjRidGNS9GOFEnaXRhbGljRidGMi9GOEY8RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GOFEnbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JC1GLzYlUSN0ckYnRjUvRjhRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiK0YnRkQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk8vJSlzdHJldGNoeUdGTy8lKnN5bW1ldHJpY0dGTy8lKGxhcmdlb3BHRk8vJS5tb3ZhYmxlbGltaXRzR0ZPLyUnYWNjZW50R0ZPLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGaG4tRiw2JUYuLUYjNiQtRi82JVEickYnRjVGQkZERkZGRA== = 0. Note that here the ball doesn't change rotation direction it will continue to do it as it did from the start i.e.CCW. Here translation is reversed. References: 1- Coriolis'sTheorie Mathematique des Effets du Jeu de Billiard, 1835, pp:51-56. 2- The problem above, presented in a different form, was taken from: Sears, Zemanski & Young's University Physics, p: 248, Problem # 9-83. It was a Challenge problem, which once I solved, helps me a great deal to understand the dynamics of the billiard ball. 3- Barger & Olsson's Classical Mechanics, pp 186-188, for the critical point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JJm1mcmFjR0YkNigtRiM2JC1JI21uR0YkNiRRIjJGJ0YvRi8tRiM2JC1GTjYkUSI1RidGL0YvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zlbi8lKWJldmVsbGVkR0Y0Ri9GK0YvR. 4- Synge & Griffith's Principles of Mechanics, pp 399-402, for the general motion of a billiard ball. Using Maple to animate the ball for the case considered : the cue hits the ball well above h=7/5R I have to consider the ball as rolling and advancing. To show the slipping I used two rays : - one blue for the real rotation and - one black to show the advancing as measured along perimeter of the ball represented as a circle. Then at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JC1GTDYlUSZmaW5hbEYnRk9GUkYvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGL0YrRi8= we get the ball rolling with a constant velocity which is the velocity of either rolling or advancing ( they are equal from that instant on ) where I used only one ray ( a black one ) to show the rolling with no slipping. Note that until we reach LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JC1GTDYlUSZmaW5hbEYnRk9GUkYvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGL0YrRi8=, friction force is operating to reduce the rolling velocity and to increase that of the advancing till both become equal than it disappears. In the present situation the rolling is well in advance from the start. The calculation below are also done to show how much rotation due to rolling and the advance get once we reach LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JC1GTDYlUSZmaW5hbEYnRk9GUkYvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGL0YrRi8=. The problem was to get an acceptable value for R i.e. around 0.025 meter= 2.5cm and v0 =6Pi. Increasing the time of rolling on the graph makes the bal smaller. This is entered in the proc as: k_vals:=seq(k*t_final*2/20,k=0..20): where the total time for the graph is double of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JC1GTDYlUSZmaW5hbEYnRk9GUkYvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGL0YrRi8=. We can make it three times or four time greater.This will, of course, gives us a smaller size ball because the x-scale being increased then the size of the ball on the y-axis will decrease. restart: with(plots): with(plottools): omega0:=5/2*v0*(h-R)/R^2;

LCQqKiMiIiYiIiMiIiJJI3YwRzYiRicsJkkiaEdGKUYnSSJSR0YpISIiRicpRixGJkYtRic=

vx:=v0+mu*g*t;

LCZJI3YwRzYiIiIiKihJI211R0YkRiVJImdHRiRGJUkidEdGJEYlRiU=

vr:=omega0*R-5/2*mu*g*t;

LCYqKiMiIiYiIiMiIiJJI3YwRzYiRicsJkkiaEdGKUYnSSJSR0YpISIiRidGLEYtRicqKkYkRidJI211R0YpRidJImdHRilGJ0kidEdGKUYnRi0=

T:=t->v0*t+1/2*mu*g*t^2;

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJkkjdjBHRiUiIiJGJEYsRiwqKiNGLCIiI0YsSSNtdUdGJUYsSSJnR0YlRiwpRiRGL0YsRixGJUYlRiU=

theta:=t->omega0*t-1/2*5/2*mu*g*t^2;

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJkknb21lZ2EwR0YlIiIiRiRGLEYsKiojIiImIiIlRixJI211R0YlRixJImdHRiVGLClGJCIiI0YsISIiRiVGJUYl

R:=0.025;h:=8/5*R;v0:=3.5*Pi;mu:=0.5;g:=10;

JCIjRCEiJA==

JCIrKysrK1MhIzY=

LCQqJiQiI04hIiIiIiJJI1BpRyUqcHJvdGVjdGVkR0YnRic=

JCIiJiEiIg==

IiM1

t_final:=evalf(2/7*(omega0*R-v0)/(mu*g));

JCIrYUVmVEohIzU=

t:=t_final;

JCIrYUVmVEohIzU=

evalf([vx,vr]);

NyQkIitpcWpjNyEiKSQiK2hxamM3RiU=

evalf([T(t),theta(t),theta(t)/T(t)]);

NyUkIitfOzUsUCEiKiQiK0ElW2sxIyEiKCQiK0tMTCRlJiEiKQ==

V:=v0+mu*g*t_final;

LCYqJiQiI04hIiIiIiJJI1BpRyUqcHJvdGVjdGVkR0YnRickIitGanpxOiEiKkYn

t_final:= 0.01*floor(100*t_final);

JCIjSiEiIw==

billiard:=proc(t0) local cir,ray,ray1,ray2,t,Dt; t:=t0; if t0 <= t_final then cir:= t-> circle([1+T(t),1],1,color=red,thickness=3): ray1:=t-> plot([[1+T(t),1],[1+T(t)+cos(-theta(t)-Pi/2),1+sin(-theta(t)-Pi/2)]] ,color=blue,thickness=3): ray2:= t->plot([[1+T(t),1],[1+T(t)+cos(-T(t)-Pi/2),1+sin(-T(t)-Pi/2)]] ,color=black,thickness=1): display(cir(t),ray1(t),ray2(t),scaling=constrained); elif t0 > t_final then Dt:= t0 - t_final; cir:= t->circle([1+T(t)+V*Dt,1],1,color=red,thickness=3): ray:= t->plot([[1+T(t)+V*Dt,1],[1+T(t)+V*Dt+cos(-T(t)-V*Dt-Pi/2) ,1+sin(-T(t)-V*Dt-Pi/2)]],color=black,thickness=1): display(cir(t),ray(t),scaling=constrained); fi; end: k_vals:=seq(k*t_final*3/20,k=0..20): to_animate:=[seq(billiard(k),k=k_vals)]: display(to_animate,insequence=true,scaling=constrained);

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