For the example illustrated above, studies have led to the function θ(t)=2t3{\displaystyle \theta (t)=2t^{3}}, where θ(t){\displaystyle \theta (t)} is the angular measure of the position of the rotation at a given time, and t{\displaystyle t} represents the time.

In this example, find the first derivative of the position function θ(t)=2t3{\displaystyle \theta (t)=2t^{3}}: ω(t)=dθdt=6t2{\displaystyle \omega (t)={\frac {d\theta }{dt}}=6t^{2}} If desired, this function could be used to calculate the angular velocity of the spinning object at any desired time t{\displaystyle t}. For this particular calculation, the angular velocity function is just an intermediate step toward finding angular acceleration.

In the previous step, you used the function for position to find the angular velocity ω(t)=6t2{\displaystyle \omega (t)=6t^{2}}. Now find the acceleration function as the derivative of ω{\displaystyle \omega }: α=dωdt=12t{\displaystyle \alpha ={\frac {d\omega }{dt}}=12t}.

For the sample problem in the illustration, suppose you know that the function for the position of the spinning object is θ(t)=2t3{\displaystyle \theta (t)=2t^{3}}, and you are asked for the object’s angular acceleration after it has been spinning for 6. 5 seconds. Use the derived formula for α{\displaystyle \alpha } and insert the information as follows: α=dωdt=12t{\displaystyle \alpha ={\frac {d\omega }{dt}}=12t} α=(12)(6. 5){\displaystyle \alpha =(12)(6. 5)} α=78. 0{\displaystyle \alpha =78. 0} Your result should be reported in units of radians per second squared. Thus, the angular acceleration for this spinning object when it has been spinning for 6. 5 seconds is 78. 0 radians per second squared.

α=ΔωΔt=final velocity-initial velocityelapsed time{\displaystyle \alpha ={\frac {\Delta \omega }{\Delta t}}={\frac {\text{final velocity-initial velocity}}{\text{elapsed time}}}} Consider a compact disc at the moment you place it in the CD player. Its initial velocity is zero. As an alternative example, suppose you know from test measurements that the wheels of a roller coaster spin at a velocity of 400 revolutions per second, which is equivalent to 2513 radians per second. To measure negative acceleration over a braking distance, you can use this measurement as the initial velocity.

A compact disc plays in the machine by rotating at an angular velocity of 160 radians per second. The roller coaster, after applying its brakes to the spinning wheels, ultimately reaches an angular velocity of zero when it stops. This will be its final angular velocity.

The owner’s manual for the CD player provides the information that the CD reaches its playing speed in 4. 0 seconds. From observations of roller coasters being tested, it has been found that they can come to a complete stop within 2. 2 seconds from when the brakes are initially applied.

For the example of the CD player, the calculation is as follows: α=ΔωΔt=final velocity-initial velocityelapsed time{\displaystyle \alpha ={\frac {\Delta \omega }{\Delta t}}={\frac {\text{final velocity-initial velocity}}{\text{elapsed time}}}} α=160−04. 0{\displaystyle \alpha ={\frac {160-0}{4. 0}}} α=1604. 0{\displaystyle \alpha ={\frac {160}{4. 0}}} α=40{\displaystyle \alpha =40} radians per second squared. For the roller coaster example, the calculation looks like this: α=ΔωΔt=final velocity-initial velocityelapsed time{\displaystyle \alpha ={\frac {\Delta \omega }{\Delta t}}={\frac {\text{final velocity-initial velocity}}{\text{elapsed time}}}} α=0−25132. 2{\displaystyle \alpha ={\frac {0-2513}{2. 2}}} α=−25132. 2{\displaystyle \alpha ={\frac {-2513}{2. 2}}} α=−1142. 3{\displaystyle \alpha =-1142. 3} radians per second squared. Note that acceleration is always going to be in units of some distance measurement “per” time squared. With angular acceleration, the distance is generally measured in radians, although you could convert that to number of rotations if you wish.

The angle that is being measured is commonly represented by θ{\displaystyle \theta }, the Greek letter theta. Positive motion is measured in a counterclockwise direction. Negative motion is measured in a clockwise direction.

One full rotation around the unit circle is said to measure 2π radians. Therefore, a half circle is π radians, and a quarter circle is π/2 radians. Sometimes it is useful to convert from radians to degrees. If you recall that a full circle is 360 degrees, you can find the conversion as follows: 360 degrees=2π radians{\displaystyle 360\ {\text{degrees}}=2\pi \ {\text{radians}}} 3602π degrees=1 radian{\displaystyle {\frac {360}{2\pi }}\ {\text{degrees}}=1\ {\text{radian}}} 57. 3 degrees=1 radian{\displaystyle 57. 3\ {\text{degrees}}=1\ {\text{radian}}} Thus, one radian is about equal to 57. 3 degrees.

People often use the word “acceleration” to mean speeding up, and “deceleration” to mean slowing down. In mathematical and physical terms, however, only the word “acceleration” is used. If the object is speeding up, the acceleration is positive. If it is slowing down, the acceleration is negative.