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When you ride a carousel, you go round and round in a circle. With each revolution you travel a linear distance one circumference. If you travel a fraction of a complete revolution, the distance you travel is the length of the arc (d) described by the angle (Q). The angular displacement is simply described by the number of  revolutions you make. The motion of planets around a sun can be described in a similar way.

Angular Displacement (q)  = # rotations = # revolutions = # degrees = # radians

If we consider our ride on the carousel it may spin slowly or quickly. While the linear velocity at any point is how fast you are going in the direction of a tangent drawn at that point, the angular velocity how many revolutions the carousel makes in a certain amount of time.

Angular Velocity (w) = # rotations / time = # revolutions / time = # degrees / time = # radians / time

w = D q /Dt.

When the carousel starts from rest it undergoes an angular acceleration until it attains some final constant angular velocity.

For Linear Motion a = Dv / Dt. Angular acceleration is essentially the same equation except we use a for a and  w for v:

Angular Acceleration (a) = change in angular velocity / change in time

a = D w / Dt

a= angular acceleration
w = angular velocity
 t = time

Finding Linear Quantities from Angular Quantities

Linear displacement, velocity and acceleration is directly related to their respective angular quantities by the following formulas:

d = rq       v = rw    a = ra

Centripetal Acceleration

We have defined acceleration as the change in velocity per change in time. Up until now, any change in velocity has been a change in the magnitude of velocity. If we consider an object  in uniform circular motion, its speed doesn't change but its direction continually changes. If the direction changes, the velocity changes and it must be accelerating. 

How do we calculate Dv ?

If we rely on our equation for linear acceleration we soon find that we've reached a dead end because the magnitude of the velocity hasn't changed. To calculate the acceleration, we must treat the velocity as a vector. The change in velocity is not determined by doing an arithmetic subtraction of the velocities but instead a vector subtraction.

A snapshot of an object in circular motion. The initial velocity is v₁. A short time later, we see the object with a velocity of v₂.

In order to calculate the acceleration, we need to determine the change in velocity, Dv. 

From what we've learned previously about acceleration, the direction of acceleration is the same as the direction of Dv. In the case of circular motion we can see from the diagram above that the direction of Dv is towards the center of the circle. In class we derived the equation for centripetal acceleration, a_c which is the acceleration an object experiences when it travels in uniform circular motion.

a_c = v²/r = w²r

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