rotational motion chapter 8. rotational variables
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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:
๐=0๐
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 Moonโs rotation in seconds?
b. What is the frequency of the Moonโs rotation in rad/s?
c. What is the linear speed of a rock on the Moonโs equator due only to the Moonโs rotation?
d. Compare this to the speed of a person at the Earthโs equator due to the Earthโs 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 โ 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
๐๐๐ข๐๐ก๐๐๐๐๐๐๐๐ค๐๐ ๐๐ก๐๐๐๐ข๐=๐๐๐๐๐๐ค๐๐ ๐ ๐ก๐๐๐๐ข๐
(๐น ๐๐๐๐ร๐)๐๐๐ค=(๐น ๐๐๐๐ร๐ )๐๐ค
๐น ๐๐๐๐๐๐๐ค=20๐ร30๐๐40๐๐
=15๐
Todayโs 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
Newtonโs Second Law for Rotation
Newtonโs 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 pulleyโs radius is 0.150 m. The pulleyโs 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 weโll 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.
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