PHYS101: Introduction to Mechanics

9a. Solve kinematics problems involving rotational motion

• Define angular velocity and the relationship between angular velocity and linear velocity.
• Define angular acceleration.
• Define kinematics.
• Calculate angular acceleration or deceleration.
• Given angular acceleration, determine the time needed for a system to accelerate or decelerate.
• Calculate the distance traveled by a rotating object

We define angular velocity (or rotational velocity) as:

$\omega = \frac{\Delta \theta}{\Delta t}$

where $\omega$ represents angular velocity.

We can relate angular velocity to linear velocity with the equation:

$v = r \omega$

where $r$ is the radius of curvature.

See a diagram of a rotating object showing the relationship between linear and angular velocity in Figure 10.3.

Angular acceleration is defined as the change in angular velocity with respect to time:

$\alpha = \frac{\Delta \omega}{\Delta t}$

where $\alpha$ represents angular acceleration.

Kinematics is the study of motion. When solving kinematics problems of rotational motion, we look at the relationships between angular velocity, linear velocity, and angular acceleration. See equations 10.17, 10.18, and 10.19 in Kinematics of Rotational Motion for kinematics equations discussed in previous sections and equations modified for rotational motion.

Example 10.1: Calculating the Angular Acceleration and Deceleration of a Bike Wheel shows a worked example of how to calculate the angular acceleration of a bike wheel. In the first part of the problem, we calculate the angular acceleration of the wheel given the change in angular velocity and time. In the second part of the problem, we calculate the time needed to stop an already spinning wheel given angular deceleration an initial velocity, using the same angular acceleration equation.

Example 10.3: Calculating the Acceleration of a Fishing Reel shows a worked example of how to calculate the kinematics of an accelerating fishing reel. Here, equation 10.19 is used to determine how the angular velocity changes with time. This result is used to calculate linear speed.

Example 10.4: Calculating the Duration When the Fishing Reel Slows Down and Stops is a worked example in which the fishing reel decelerates. Using equation 10.19, we solve for time rather than angular velocity.

Review for additional worked examples of these types of problems in Example 10.5: Calculating the Slow Acceleration of Trains and their Wheels and Example 10.6: Calculating the Distance Traveled by a Fly on the Edge of a Microwave Oven Plate.

9b. Solve dynamics problems involving rotational motion

• Define dynamics.
• List the steps for solving dynamics problems for rotational motion.

The study of dynamics involves calculations of how force and mass impact an object's motion. When solving dynamics problems, first we need to identify the system and draw a free body diagram of all the forces acting upon the system. Once the forces acting upon the system are defined, we can use the following torque equation and angular acceleration equations to solve the problem:

$\mathrm{net\ }\tau=I\alpha$, and $\alpha=\frac{\mathrm{net\ }\tau}{I}$

where $I$ is the moment of inertia.

Example 10.7: Calculating the Effect of Mass Distribution on a Merry-Go-Round shows a worked example of using the above equations to determine the angular acceleration of a person pushing a merry-go-round. Here, the first step is to calculate torque. The next step is to calculate the moment of inertia. Lastly, torque and moment of inertia are used to calculate the angular acceleration on the merry-go-round.

9c. Define rotational inertia

• Define rotational inertia.
• Define moment of inertia.

We define rotational inertia as $mr^2$, where $m$ is the mass of the object being rotated and $r$ is the radius from the pivot point to the end of the mass.

Review a diagram of an object rotating on a frictionless table in Figure 10.11. We can see the radius from the center of the table (the pivot point) and the mass at the end of the radius. 

The moment of inertia, $I$, is the sum of all the rotational inertia acting upon an object. We can write the following summation:

$I=\sum mr^2$

9d. Compare and contrast the dynamics of linear and rotational motion

•  How are the definitions of linear and rotational motion related to each other? 

Rotational velocity is related to linear velocity in the following equation:

$v=r\omega$

where $r$ is the radius of curvature.

9e. Apply energy concepts to rotational motion

• Define work for rotational motion.
• Define rotational kinetic energy.

For rotational motion to occur, work must be done. However, we cannot use the simple definitions of work that we used earlier in this course, because we now must account for rotational motion.

Review a diagram of a spinning disk in Figure 10.15.

For the disk to spin, work must be done on the disk. The force acting upon the disk must be perpendicular to the radius of the disk, which we know is torque. We also know torque is related to moment of inertia. We can relate the work done on the disk to moment of inertia using the following equations:

$\mathrm{net\ }\omega=\left ( \mathrm{net\ }\tau \right )\theta=I\alpha\theta$

We can write an equation for the rotational kinetic energy (the energy of rotational motion) as:

$\mathrm{KE_{rot}}=\frac{1}{2}I\omega^2$

Example 10.8: Calculating the Work and Energy for Spinning a Grindstone is a worked example for calculating the net work for a rotating disk using the above work equation. In the second part of the example, the rotational velocity is determined using the equation for rotational acceleration and moment of inertia. Lastly, the rotational kinetic energy is calculated using the above equation.

9f. Give several everyday examples of conservation of angular momentum

• Define angular momentum.
• Describe everyday examples of conservation of angular momentum.

We define angular momentum as $L=I\omega$.

It is similar to the momentum defined for linear motion. As such, angular momentum in a system is conserved in the same way that linear momentum is conserved. Therefore, we can say that $L=L'$, where $L$ is the initial angular momentum in a system and $L'$ is the final angular momentum in the system. We can also write this as:

$I\omega=I'\omega'$

We see conservation of angular momentum in many everyday examples. One example is the figure skater spinning in Figure 10.23. In the first picture, the figure skater is spinning with her arms out on a frictionless ice surface. In the second picture, she pulls her arms in, and her rotational velocity increases. When the figure skater pulls in her arms, she lowers her moment of inertia. Because angular momentum is conserved, because her moment of inertia decreases, her angular velocity must therefore increase.

Unit 9 Vocabulary

• Angular acceleration
• Angular momentum
• Angular velocity
• Dynamics
• Kinematics
• Linear velocity
• Moment of inertia
• Rotational inertia
• Rotational kinetic energy