Rotational MotionChapter 8

Rotational Variables Angular displacement is the

change in the angle or .

Angular velocity is the angular displacement divided by the time required to make the rotation:

Angular acceleration is the change in angular velocity divided by the time required to make that change:

Angular frequency is the number of revolutions per second:

𝜃=0𝜃

is measured in radians; there are radians in a circle. The positive direction is counterclockwise.

Comparison between Linear and Rotational Variables

QuantityLinear

VariablesRotational Variables

Relationship

Displacement

Velocity

Acceleration

Example Problems 8-5. A movie lasts two hours. During that time, what is the

angular displacement of each of the following: a. the hour hand of a watch

b. the minute hand of a watch.

8-6. The Moon rotates once on its axis in 27.3 days. Its radius is 1.74 X 106 meters. a. What is the period of the Moon’s rotation in seconds?

b. What is the frequency of the Moon’s rotation in rad/s?

c. What is the linear speed of a rock on the Moon’s equator due only to the Moon’s rotation?

d. Compare this to the speed of a person at the Earth’s equator due to the Earth’s rotation

Example Problems - 2

8-9. In the spin cycle of a clothes washer, the drum turns at 635 rev/min. If the lid of the washer is opened, the lid of the washer is opened, the washer turns off. If the drum requires 8.0 s to come to a stop, what is the angular acceleration of the drum?

8-10. A CD-ROM has a spiral track that starts 2.7 cm from the center of the disk and end 5.5 cm from the center. The disk drive must turn the disk so that the linear velocity of the track is a constant 1.4 m/s. Find the following: a. the angular velocity of the disk for the start of the track.

b. the angular velocity of the disk for the end of the track.

c. the angular acceleration if the track plays for 76 min.

Torque

Lever arm – the perpendicular distance from the axis of rotation to the point where the force is exerted.

where is the distance from the axis of rotation to the point where the force is applied, and is the angle between the force and the radius of rotation.

Torque is a measure of how effectively a force causes a rotation and is the product of the force and the lever arm:

The units of torque are newton-meters (.

Torque

The simple case is when the force is perpendicular to the lever arm. Then:

Balanced Torques

Torques are balanced if the sum of torques that tend to produce a clockwise rotation is equal to the sum of the torques that produces a counterclockwise rotation. If torques are balanced there is no rotation.

30 cm

20 N? N

40 cm

𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒𝑡𝑜𝑟𝑞𝑢𝑒=𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑡𝑜𝑟𝑞𝑢𝑒

(𝐹 𝑝𝑒𝑟𝑝×𝑑)𝑐𝑐𝑤=(𝐹 𝑝𝑒𝑟𝑝×𝑑 )𝑐𝑤

𝐹 𝑝𝑒𝑟𝑝𝑐𝑐𝑤=20𝑁×30𝑐𝑚40𝑐𝑚

=15𝑁

Today’s Activities

Center of Mass Builder

Balancing Act simulation

Torque Worksheet

Useful video: Solving Torque Problems

Moment of Inertia

In the same way that inertia is the resistance of an object to a change of motion, the moment of inertia ()of an object is the resistance of an object to a rotation.

For a point mass located a distance from the axis of rotation, its moment of inertia is:

In general the moment of inertia depends on how the mass of the object is distributed and on the location of the rotational axis. For example, for a long uniform rod of length

if the axis of rotation is through the center of the bar

if the axis of rotation is through one end of the bar

Examples of Rotational Inertia

Newton’s Second Law for Rotation

Newton’s Second Law for Rotational Motion The angular acceleration of an object is equal to the net

torque on the object divided by the moment of inertia.

Example Problems

8-29. A disk with a moment of inertia of 0.26 kg-m2 is attached to a smaller disk mounted on the same axle. The smaller disk has a diameter of 0.180 m and a mass of 2.5 kg. A strap is wrapped around the smaller disk as shown in Figure 8-10. Find the force needed to give this system an angular acceleration of 2.57 rad/s2.

8-34. A rope is wrapped around a pulley and pulled with a force of 13.0 N. The pulley’s radius is 0.150 m. The pulley’s rotational speed goes from 0.0 rev/min to 14.0 rev/min in 4.50 s. What is the moment of inertia of the pulley?

Center of Gravity/Center of Mass

We can treat an entire object as if all of its mass were concentrated at a single point.

That point is known as the center of mass. It is the weighted average of the location of mass in an object.

For two particles of mass and located at positions and respectively, the center of mass is located at:

Center of Mass Simulator

Boys on Skates

The center of gravity is the point at which the force of gravity acts upon an object were a point particle with all of its mass at that point. For the problems we’ll consider, center of mass and center of gravity are the same.

Finding the Center of Mass

Center of Mass Applications

Fosbury Flop

Fosbury Flop

Science of NFL Football

Circus Physics

Pluto and Charon

Example Problem

8-37. A 7.3 kg ladder, 1.92 m long, rests on two sawhorses as shown. Sawhorse A is located 0.30 m from the end, and sawhorse B is located 0.45 m from the other end. Choose the axis of rotation to be the center of mass of the ladder.A. What are the torques acting on the ladder?B. Write the equation for rotational equilibrium.C. Solve the equation for FA in terms of Fg.D. How would the forces exerted by the two sawhorses change if A were moved very close to, but not under the center of mass?

0.30 m 0.66 m 0.51 m 0.45 m

A B

CoM

Stability and Equilibrium

An object balanced so that any displacement lowers its center of gravity is in unstable equilibrium.

An object balanced so that any displacement raises its center of gravity is in stable equilibrium.

An object that is balanced so that any displacement neither raises or lowers its center of gravity is in neutral equilibrium.

An object with a low center of gravity is usually more stable than an object with a nigh center of gravity.

If the center of gravity extends beyond the support base of an object, the object will topple.