Rotational Motion
Learn how to describe and measure rotational motion.
Learn how torque changes rotational velocity.
Define center of mass and the conditions for equilibrium.
Chapter
8
In this chapter you will:
Table of Contents
Chapter 8: Rotational Motion
Section 8.1: Describing Rotational Motion
Section 8.2: Rotational Dynamics
Section 8.3: Equilibrium
Chapter
8
Assignments:
Read Chapter 8.
Study Guide 8 due before the Chapter Test.
HW 8.A: p.223: 72-77.
HW 8.B: p.224: 81,82,84. p.225: 91,97.
HW 8.C: Handout
Describing Rotational Motion
Describe angular displacement.
Calculate angular velocity.
Calculate angular acceleration.
Solve problems involving rotational motion.
In this section you will:
Section
8.1
One complete revolution is equal to 2π radians, so
360 degrees = 2 π radians
Describing Rotational Motion
A fraction of one revolution can be measured in grads, degrees, or radians.
Describing Rotational Motion
Section
8.1
A grad is 1/400 of a revolution.
A degree is 1/360 of a revolution.
The radian is defined as ½ π of a revolution. The abbreviation of radian is ‘rad’. The distance around the circle is 2π
Length = d
For rotation through an angle, θ, a point at a distance, r, from the
center moves a distance given by d = r
The Greek letter theta, θ, is used to represent the angle of revolution.
The counterclockwise rotation is designated as positive, while clockwise is negative.
Angular Displacement
Section
8.1
As an object rotates, the change in the angle is called angular displacement.
Describing Rotational Motion
Velocity is displacement divided by the time taken to make the displacement.
The angular velocity of an object is angular displacement divided by the time required to make the displacement.
Angular Velocity
Section
8.1
The angular velocity of an object is given by: = t
Here angular velocity is represented by the Greek letter omega, ω.
The angular velocity is equal to the angular displacement divided by the time required to make the rotation.
Angular velocity is measured in rad/s.
Describing Rotational Motion
For Earth, ωE = (2π rad)/(24.0 h)(3600 s/h) = 7.27×10─5 rad/s.
Angular Velocity
Section
8.1
In the same way that counterclockwise rotation produces positive angular displacement, it also results in positive angular velocity.
If an object’s angular velocity is ω, then the linear velocity of a point a
distance, r, from the axis of rotation is given by v = r ω.
The speed at which an object on Earth’s equator moves as a result of Earth’s rotation is given by v = r ω = (6.38×106 m) (7.27×10─5 rad/s) = 464 m/s.
Earth is an example of a rotating, rigid object. Even though different points on Earth rotate different distances in each revolution, all points rotate through the same angle.
Describing Rotational Motion
Angular acceleration is defined as the change in angular velocity divided by the time required to make that change.
Angular Acceleration
Section
8.1
Angular acceleration is measured in rad/s2.
If the change in angular velocity is positive, then the angular acceleration also is positive.
The linear acceleration of a point at a distance, r, from the axis of an
object with angular acceleration, α, is given by a = r α .
The angular acceleration, α, is represented by the following equation:
α = t
Describing Rotational Motion
A summary of linear and angular relationships.
Angular Acceleration
Section
8.1
p.200: Practice Problems: 1,2. Section Review: 5,7-10.
HW 8.A: p.223: 72-77.
Section Check
What is the angular velocity of the minute hand of a clock?
Question 1
Section
8.1
A.
B.
C.
D.
Section Check
Answer: B
Answer 1
Section
8.1
Reason: Angular velocity is equal to the angular displacement divided by the time required to complete one rotation.
In one minute, the minute hand of a clock completes one rotation. Therefore, = 2π rad.
Therefore,
Section Check
When a machine is switched on, the angular velocity of the motor increases by 10 rad/s for the first 10 seconds before it starts rotating with full speed. What is the angular acceleration of the machine in the first 10 seconds?
Question 2
Section
8.1
A. π rad/s2
B. 1 rad/s2
C. 100π rad/s2
D. 100 rad/s2
Section Check
Answer: B
Answer 2
Section
8.1
Reason: Angular acceleration is equal to the change in angular velocity divided by the time required to make that change.
Section Check
When a fan performing 10 revolutions per second is switched off, it comes to rest after 10 seconds. Calculate the average angular acceleration of the fan after it was switched off.
Question 3
Section
8.1
A. 1 rad/s2
B. 2π rad/s2
C. π rad/s2
D. 10 rad/s2
Section Check
Answer: B
Answer 3
Section
8.1
Reason: Angular displacement of any rotating object in one revolution is 2π rad.
Since the fan is performing 10 revolution per second, its angular velocity = 2π × 10 = 20π rad/s.
Angular acceleration is equal to the change in angular velocity divided by the time required to make that change.
