Uniform Circular Motion
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, respectively, to the center of rotation:
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, every acceleration is the result of a corresponding force, and hence the term centripetal force is often used.
Here are a few examples of centripetal forces that can result in circular motion:
When a weight is rotated at the end of a string, the string's tension provides the centripetal force.
An orbiting celestial body such as the Earth or Moon stays in orbit due to gravity, a centripetal force.
When a car turns a corner, the force of friction between the tires and the ground is a centripetal force.
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):
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 = r⋅ω .
This formula can also be derived in a different way, using vectors. If you take the position vector to be
where ur=cosθ, sinθ, then the (linear) velocity is given by
where uθ=−sinθ, cosθ. Then the speed is the magnitude of the velocity:
v = v = r⋅ω.
Finally, you can compute the (centripetal) acceleration in the same way:
a=ⅆvⅆt=−r⋅ur⋅ⅆθⅆt⋅ω+r⋅uθ⋅ⅆωⅆt = −r⋅ω2.
The magnitude of the centripetal acceleration is thus:
a = a = r⋅ω2 = v⋅ω = v2r
and the associated centripetal force is:
Fc = ma = mv2r.
Adjust the angular velocity w. Observe that the centripetal force and acceleration act toward the center of the circle
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