10.3 Dynamics of rotational motion: rotational inertia

Learning objectives

By the end of this section, you will be able to:

• Understand the relationship between force, mass, and acceleration.
• Study the turning effect of force.
• Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration.

The information presented in this section supports the following AP® learning objectives and science practices:

• 4.D.1.1 The student is able to describe a representation and use it to analyze a situation in which several forces exerted on a rotating system of rigidly connected objects change the angular velocity and angular momentum of the system. (S.P. 1.2, 1.4)
• 4.D.1.2 The student is able to plan data collection strategies designed to establish that torque, angular velocity, angular acceleration, and angular momentum can be predicted accurately when the variables are treated as being clockwise or counterclockwise with respect to a well-defined axis of rotation, and refine the research question based on the examination of data. (S.P. 3.2, 4.1, 5.1, 5.3)
• 5.E.2.1 The student is able to describe or calculate the angular momentum and rotational inertia of a system in terms of the locations and velocities of objects that make up the system. Students are expected to do qualitative reasoning with compound objects. Students are expected to do calculations with a fixed set of extended objects and point masses. (S.P. 2.2)

If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in [link] . In fact, your intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely proportional to mass. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton's second law of motion. There are, in fact, precise rotational analogs to both force and mass.

To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force $F$ on a point mass $m$ that is at a distance $r$ from a pivot point, as shown in [link] . Because the force is perpendicular to $r$ , an acceleration $a=\frac{F}{m}$ is obtained in the direction of $F$ . We can rearrange this equation such that $F=\text{ma}$ and then look for ways to relate this expression to expressions for rotational quantities. We note that $a=\mathrm{r\alpha }$ , and we substitute this expression into $F=\text{ma}$ , yielding

Recall that torque    is the turning effectiveness of a force. In this case, because $\mathbf{\text{F}}$ is perpendicular to $r$ , torque is simply $\tau =\mathrm{Fr}$ . So, if we multiply both sides of the equation above by $r$ , we get torque on the left-hand side. That is,