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,

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

or

$\tau=m r^{2} \alpha$

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.

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 $r$.

Source: Rice University, https://openstax.org/books/college-physics/pages/10-3-dynamics-of-rotational-motion-rotational-inertia
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