Rotational Inertia

As you read, pay attention to the 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.

When solving dynamics problems, we first 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 torque equation and angular acceleration equations to solve the problem:  \tau_{net}=I\alpha , where  I is the moment of inertia,  \tau is torque, and  \alpha is the rotational acceleration due to the torque.

Example 10.7 shows how to use these 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. Finally, torque and moment of inertia are used to calculate the angular acceleration on the merry-go-round.

Dynamics of Rotational Motion: Rotational Inertia

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 Figure 10.10. 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.

Figure 10.10 Force is required to spin the bike wheel. The greater the force, the greater the angular acceleration produced. The more massive the wheel, the smaller the angular acceleration. If you push on a spoke closer to the axle, the angular acceleration will be smaller.

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 Figure 10.11. 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=m a and then look for ways to relate this expression to expressions for rotational quantities. We note that a=r \alpha, and we substitute this expression into F=m a, yielding

F=m r \alpha.

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

r F=m r^{2} \alpha.


\tau=m r^{2} \alpha

This last equation is the rotational analog of Newton's second law (F=m a), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and m r^{2} is analogous to mass (or inertia). The quantity m r^{2} is called the rotational inertia or moment of inertia of a point mass m a distance r from the center of rotation.

The given figure shows an object of mass m, kept on a horizontal frictionless table, attached to a pivot point, which is in the center of the table, by a cord that supplies centripetal force. A force F is applied to the object perpendicular to the radius r, which is indicated by a red arrow tangential to the circle, causing the object to move in counterclockwise direcion.

Figure 10.11 An object is supported by a horizontal frictionless table and is attached to a pivot point by a cord that supplies centripetal force. A force F is applied to the object perpendicular to the radius r, causing it to accelerate about the pivot point. The force is kept perpendicular to r.

Making Connections

Dynamics for rotational motion is completely analogous to linear or translational dynamics. Dynamics is concerned with force and mass and their effects on motion. For rotational motion, we will find direct analogs to force and mass that behave just as we would expect from our earlier experiences.


Source: Rice University,
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