8.5 Dynamics of rotational motion: rotational inertia  (Page 4/8)

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Torque is the analog of force and moment of inertia is the analog of mass. Force and mass are physical quantities that depend on only one factor. For example, mass is related solely to the numbers of atoms of various types in an object. Are torque and moment of inertia similarly simple?

No. Torque depends on three factors: force magnitude, force direction, and point of application. Moment of inertia depends on both mass and its distribution relative to the axis of rotation. So, while the analogies are precise, these rotational quantities depend on more factors.

Section summary

• The farther the force is applied from the pivot, the greater is the angular acceleration; angular acceleration is inversely proportional to mass.
• If we exert a force $F$ on a point mass $m$ that is at a distance $r$ from a pivot point and because the force is perpendicular to $r$ , an acceleration $\text{a = F/m}$ is obtained in the direction of $F$ . We can rearrange this equation such that
$\mathrm{F = ma}\text{,}$

and then look for ways to relate this expression to expressions for rotational quantities. We note that $\mathrm{a = r\alpha }$ , and we substitute this expression into $\mathrm{F=ma}$ , yielding

$\mathrm{F=mr\alpha }$
• Torque is the turning effectiveness of a force. In this case, because $F$ is perpendicular to $r$ , torque is simply $\tau =\mathit{rF}$ . If we multiply both sides of the equation above by $r$ , we get torque on the left-hand side. That is,
$\text{rF}={\text{mr}}^{2}\alpha$

or

$\tau ={\text{mr}}^{2}\alpha \text{.}$
• The moment of inertia $I$ of an object is the sum of ${\text{MR}}^{2}$ for all the point masses of which it is composed. That is,
$I=\sum {\text{mr}}^{2}\text{.}$
• The general relationship among torque, moment of inertia, and angular acceleration is
$\tau =\mathrm{I\alpha }$

or

$\begin{array}{}\alpha =\frac{\text{net τ}}{I}\cdot \\ \end{array}$

Conceptual questions

The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is ${\mathit{ML}}^{2}\text{/3}$ . Why is this moment of inertia greater than it would be if you spun a point mass $M$ at the location of the center of mass of the rod (at $L/2$ )? (That would be ${\mathit{ML}}^{2}\text{/4}$ .)

Why is the moment of inertia of a hoop that has a mass $M$ and a radius $R$ greater than the moment of inertia of a disk that has the same mass and radius? Why is the moment of inertia of a spherical shell that has a mass $M$ and a radius $R$ greater than that of a solid sphere that has the same mass and radius?

Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque.

While reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame? The image shows a side view of a racing bicycle. Can you see evidence in the design of the wheels on this racing bicycle that their moment of inertia has been purposely reduced? (credit: Jesús Rodriguez)

A ball slides up a frictionless ramp. It is then rolled without slipping and with the same initial velocity up another frictionless ramp (with the same slope angle). In which case does it reach a greater height, and why?

Problems&Exercises

This problem considers additional aspects of example Calculating the Effect of Mass Distribution on a Merry-Go-Round . (a) How long does it take the father to give the merry-go-round and child an angular velocity of 1.50 rad/s? (b) How many revolutions must he go through to generate this velocity? (c) If he exerts a slowing force of 300 N at a radius of 1.35 m, how long would it take him to stop them?

(a) 0.338 s

(b) 0.0403 rev

(c) 0.313 s

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in general
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