In this article (and video above), we take a deep dive into the world of rotational kinematics, and explore the significance of using the 4 constant angular acceleration equations to solve complex problems and unravel the mysteries of rotating objects.

**Question:**

The disk presented in the figure accelerates uniformly from rest at a rate of 2 rad/s^2 for 4 seconds. It then maintains a constant angular velocity for 4 seconds before decelerating uniformly at a rate of 5 rad/s^2 until it comes to rest.

What is the total angular displacement of the disk?

**Problem C****ontext****:**

### Kinematics of particles

Particle kinematics is the study of the movement of particles, without considering the forces that cause this movement.

Be careful if acceleration isn’t a simple number. You’ll need to use a different method than what we’ve learned so far. But let’s try these basic ideas on our problem now.

**The four basic kinematics equations
(constant acceleration equations):**

**The four basic kinematics equations
(constant **

**angular**

**acceleration equations):**

To figure out how far the disk spins, we need to break its movement into three parts:

- It speeds up for 4 seconds.
- It spins at a steady speed for another 4 seconds.
- It slows down until it stops.

By adding up how far it spins in each part, we can find out the total distance it spins.

1. **Uniform acceleration from rest**

**2. Constant angular velocity**

**3. Uniform deceleration to rest**

**Phase 1 – Uniform acceleration from rest**

**1. Calculate angular displacement θ _{1}:**

We’ve labeled the distance the disk moves in each part as theta 1, 2, and 3. This is what we’ll be figuring out.

During the first phase, the disk starts from a standstill and spins faster for 4 seconds. We know how quickly it speeds up, so we can figure out how much it spins in total during those 4 seconds.

By substituting these known quantities into the equation of motion, we find that the disk experiences a displacement of 16 radians.

**Phase 2 – Constant angular velocity**

During the second phase, the disk spins at a steady speed, but we don’t know how fast. To figure this out, we need to know how fast it’s spinning at the end of the first phase when it stops speeding up. This speed will stay the same for the whole second phase.

By evaluating the list of available equations, we note that we have all the necessary information to use equation 1.

**2. Calculate angular velocity ω_1:**

After 4 seconds, the disk is spinning at 8 times per second. Now we can move on to phase 2.

Our disk is now spinning at a constant angular velocity, meaning its angular acceleration must be zero. With this information available, we notice that we can directly calculate theta2 using equation 3

**Phase 3 – ****Uniform deceleration to rest**

In the final phase, the disk slows down until it stops. We know how fast it was spinning from before and how quickly it slows down. We can use either of two methods to figure out how much it spins while stopping.

**1. Calculate angular displacement θ _{2}:**

We enter the numbers into the equation again and find that the disk spins 32 times during phase 2. Now we just need to figure out what happens in phase 3.

We realize that we don’t yet know the duration of this phase.

But since we know the disk comes to a stop, we also know that its final angular velocity would be zero. Equation 1 can then be used to find that it takes the disk 1.6 seconds to come to rest.

And with the phase duration known, we can now use either equation 2 or 3 to find that the angular displacement during phase 3 is 6.4 rad.

**2. Calculate total angular displacement:**

Finally, the total angular displacement of the disk can be found as 54.4 radians through simple addition.

**Answer:**

The correct answer is B.

**Final Remarks:**

1. Differentiate between acceleration and deceleration – sign convention

2. Only apply these formulas where constant acceleration is involved

3. Take advantage of the plug-and-play nature of the constant acceleration equations

**This Episode Is Brought to You by PPI**

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I hope you found this article helpful. In upcoming articles, I will solve some more PE exam practice problems and answer other questions from our subscribers. Pass the PE Exam videos will publish weekly, so be sure to click the subscribe button so you don’t miss something that could make a substantial difference in your exam result.

Lastly, I encourage you to ask questions in the comments of this video, or on this page and I’ll read and respond to them in future videos. So, if there’s a specific topic you want me to cover or answer, we have you covered.

I’ll see you next week… on Pass the FE Exam

Anthony Fasano, P.E.

Engineering Management Institute

Author of Engineer Your Own Success

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