CHAPTER 8Rotational Kinematics

Acknowledgements © Mark Lesmeister/Pearland ISD Selected questions © 2010 Pearson Education

Inc. Selected graphics and questions from Cutnell and

Johnson, Physics 9e: Instructor’s Resource Site, © 2015 John Wiley and Sons.

Consider the motion of a rigid body about a fixed axis

Discovery Lab: Rotational Motion

Go to this page on your laptop or computer: http://phet.colorado.edu/en/simulation/rotation

Click on Run Now (or Download if it is your computer and you want to have a copy of the app.)

Experiment with different angular velocities on the intro tab.

Rotational Quantities

When an object spins, it is said to undergo rotational motion

The axis of rotation is the line about which the rotation occurs.

It is difficult to describe the motion of a point moving in a circle using only linear quantities because the direction of motion in a circular path is constantly changing. For this reason, circular motion is described in terms of the angle through which the point on an object moves.

Angles can be measured in radians◦Radians – an angle whose arc length is

equal to its radius◦ In general, any angle θ measured in

radians is defined by the following:◦ s = arc length r = length of radius Θ = angle of rotation

Rotational Quantities

Rotational Quantities

The angle of 360o is one revolution. (1 rev = 360o) One revolution is equal to the circumference of the

circle of rotation. Circumference is 2pr Therefore:

so,

1 rev = 360o = 2p rad Converting angular displacement to radians:

Converting radians to degrees:

8.1 Rotational Motion and Angular Displacement

r

s

Radius

length Arcradians)(in

For a full revolution:

360rad 2 rad 22

r

r

8.1 Rotational Motion and Angular Displacement

The angle through which the object rotates is called theangular displacement.

o

Angular displacement (Dq)

Angular displacement describes how far an object has rotated

It is defined as:◦ The angle through which a point, line, or body is

rotated in a specified direction and about a specified axis (in radians)

◦ Angular displacement =

◦ Units: radiansCounterclockwise (CCW) rotation is

considered (+)Clockwise (CW) rotation is considered (-)

Δθ = θ2 – θ1

Example problem #1

John Glenn, in 1962, circled the earth 3 times in less than 5 hours. If his distance from the center of the earth was 6560 km, what arc length did he travel through? (answer in Km)

G:

U:

E:

S:

S:

x

Example problem #2

While riding on a carousel that is rotating clockwise, a child travels through an arc length of 11.5 m. If the child’s angular displacement is 165o,what is the radius of the carousel?

G: =

U:

E:

S: r =

S: 3.99 m

Angular Displacement Example – Try this

Two people ride on a carousel. One rides on a horse located 5 meters from the center. The other rides on a swan located 3 meters from the center.

When the carousel goes around ¼ of a revolution, how far does each person travel?

r s

Horse: 7.85 m

Swan: 4.71 m

4

2 Where:

Angular speed (Ѡ)The rate at which a body rotates about

an axis, which is the rate of change of angular position. expressed in: rad/sec or per sec. (T-1)

Average angular speed =

Example problem #3

In 1975, an ultra-fast centrifuge attained an average angular speed of 2.65 x 104 rad/sec. What was the angular displacement after 1.5 sec?

G: Ѡ = 2.65 x 104 , t = 1.5 s

U: Δθ = ?

E: S: Δθ = Ѡavg t

= 2.65 x 104 x 1.5S: 3.98 x 104 radians

Example problem #4

A child at an ice cream parlor spins on a stool. The child turns counterclockwise with an average angular speed of 4.0 rad/s. In what time interval will the child’s feet have an angular displacement of 8.0π rad?

G: Ѡ = 4.0 rads/s, Δθ = 8.0 π radU: E: S: t = = =2.0 π sS:

Angular acceleration (α)

The rate of change of angular speed, increase or decrease in rotational speed of the particle, expressed in rad/s/s or T-2.

or Angular acceleration =

◦wf = final angular speed

◦wo = initial angular speed

◦ = a angular acceleration◦ Dt = elapsed time

t

Example problem #5A car’s tire rotates at an initial angular speed

of 21.5 rad/s. The driver accelerates, and after 3.5 s the tire’s angular speed is 28.0 rad/s. What is the tire’s average angular acceleration during the 3.5 s time interval?

G: Ѡo = 21.5 rad/s, Ѡf = 28.0 rad/s, t = 3.5s

U: α = ?E: S: S: 1.86 rad/s2

Equations of motion for constant acceleration

Straight line motionRotation about a fixed axis

tvvx

xavv

atvv

attvx

)(

2

021

20

2

0

221

0

t

t

tt

)(

2

021

20

2

0

221

0

In comparing angular and linear quantities, they are similar. you may write this at the bottom of your notes you will need it

Analogies Between Linear and Rotational Motion:

SECTION ASSIGNMENT

SECTION 2:Tangential and centripetal acceleration

Tangential speed

Objects in circular motion have a tangential speedTangent – a line that lies in a plane of circle that

intersects at one point.Tangential speed (vt) is the instantaneous linear

speed of an object directed along the tangent to the object’s circular path.◦ Tangential speed = (distance from the axis) X

(angular speed)

= instantaneous angular speed because the time interval is too small

This equation is valid only when ω is measured in rad/s

Tangent line

Example problem #1The radius of a CD is 0.06 m. If a microbe

riding on the disc’s rim has a tangential speed of 1.88 m/s, what is the microbe’s angular speed?

G: U: ω = ?E: S: S: 31.33 rad/s

Tangential acceleration

Tangential acceleration (at) is defined as the instantaneous linear acceleration of an object directed along the tangent to the objects circular path

Tangential acceleration = (distance from the axis) X (angular acceleration)

◦ α = is the instantaneous angular acceleration◦ In this equation, you must use the unit

radians to be valid

Total acceleration

The total acceleration is the vector sum of centripetal acceleration, which points to the center of the circle, and tangential acceleration, which is tangent to the circle.

at

aC

22tCtotal aaa

Write this equation at the bottom

of your page

Example problem #2

A spinning ride at a carnival has an angular acceleration of 0.50 rad/s2. How far from the center is a rider who has a tangential acceleration of 3.3 m/s2?

G: U: r = ?E: S: S: 6.6 m

◦Centripetal acceleration can be calculated using angular speed as well:

Ex.-swinging a stopper above your head

Centripetal Acceleration in Angular Form

raC2

r

va tC

2

r

raC

2)(

Write this at the bottom

of your page

SECTION ASSIGNMENT

Due today!!!