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Uniform Circular Motion

Main Concept

Angular velocity is a quantity representing how fast an object is moving around a given fixed point called the center of rotation. When considering motion in three-dimensions, the center of rotation is no longer a point, but an axis. More precisely, the angular velocity measures the rate of change of the angle formed between the lines joining the object's initial and final positions.


When an object moves along a circle at constant speed, v, its angular velocity about the circle's center is also constant, and the object is said to move with uniform circular motion. Although the object's speed is constant, its direction of motion keeps changing, being always tangent to the circular path. Therefore, the linear velocity, v, is not constant. The acceleration required to keep the object on the circular path is called the centripetal acceleration and is directed towards the center of the circle. According to Newton's second law, each acceleration has a corresponding force, and hence the term centripetal force is often used. The force can be supplied by many things, for instance when a car turns a corner, it is the force of friction between the tires and the ground that causes the centripetal acceleration. At the Large Hadron Collider in CERN, fundamental particles are accelerated in a large circle using magnetic forces.

 

Equations of motion

The arc length s of a circle is the product of the circle's radius r and the angle θ (as illustrated earlier):

 

s= rθ,

where θ is measured in radians.

 

Taking the derivative with respect to time, you can determine that the speed of an object in uniform circular motion is the product of the radius and the angular velocity:

 

ⅆsⅆt= r ⅆθⅆt,

v  = rω .

 

This formula can also be derived in a different way, using vectors. If you take the position vector to be:

rur ,

 

 

where ur=cosθ, sinθ, then the (linear) velocity is given by:

v=ⅆrⅆt=ruθⅆθⅆt ,

 

 

where uθ=sinθ, cosθ. Then the speed is the magnitude of the velocity:

v = v = rω

 

Finally, you can compute the (centripetal) acceleration in the same way:

a=ⅆvⅆt=rurⅆθⅆtω+ruθⅆωⅆt = rω2

 

The magnitude of the centripetal acceleration is thus:

a = a = rω2 = vω = v2r

 

and the associated centripetal force is:

Fc = ma = mv2r

 

Adjust the angular velocity ω. Observe that the centripetal force and acceleration act toward the center of the circle

 

  

   ω =

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