# 10.3 Dynamics of rotational motion: rotational inertia  (Page 5/9)

 Page 5 / 9

## Test prep for ap courses

A piece of wood can be carved by spinning it on a motorized lathe and holding a sharp chisel to the edge of the wood as it spins. How does the angular velocity of a piece of wood with a radius of 0.2 m spinning on a lathe change when a chisel is held to the wood's edge with a force of 50 N?

1. It increases by 0.1 N•m multiplied by the moment of inertia of the wood.
2. It decreases by 0.1 N•m divided by the moment of inertia of the wood-and-lathe system.
3. It decreases by 0.1 N•m multiplied by the moment of inertia of the wood.
4. It decreases by 0.1 m/s 2 .

(b)

A Ferris wheel is loaded with people in the chairs at the following positions: 4 o'clock, 1 o'clock, 9 o'clock, and 6 o'clock. As the wheel begins to turn, what forces are acting on the system? How will each force affect the angular velocity and angular momentum?

A lever is placed on a fulcrum. A rock is placed on the left end of the lever and a downward (clockwise) force is applied to the right end of the lever. What measurements would be most effective to help you determine the angular momentum of the system? (Assume the lever itself has negligible mass.)

1. the angular velocity and mass of the rock
2. the angular velocity and mass of the rock, and the radius of the lever
3. the velocity of the force, the radius of the lever, and the mass of the rock
4. the mass of the rock, the length of the lever on both sides of the fulcrum, and the force applied on the right side of the lever

(d)

You can use the following setup to determine angular acceleration and angular momentum: A lever is placed on a fulcrum. A rock is placed on the left end of the lever and a known downward (clockwise) force is applied to the right end of the lever. What calculations would you perform? How would you account for gravity in your calculations?

Consider two sizes of disk, both of mass M . One size of disk has radius R ; the other has radius 2 R . System A consists of two of the larger disks rigidly connected to each other with a common axis of rotation. System B consists of one of the larger disks and a number of the smaller disks rigidly connected with a common axis of rotation. If the moment of inertia for system A equals the moment of inertia for system B, how many of the smaller disks are in system B?

1. 1
2. 2
3. 3
4. 4

(d)

You are given a thin rod of length 1.0 m and mass 2.0 kg, a small lead weight of 0.50 kg, and a not-so-small lead weight of 1.0 kg. The rod has three holes, one in each end and one through the middle, which may either hold a pivot point or one of the small lead weights.

How do you arrange these objects so that the resulting system has the maximum possible moment of inertia? What is that moment of inertia?

## 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

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

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