PHYSICS CHAPTER 4

1

CHAPTER 5: CHAPTER 5:

Rotational KinematicsRotational Kinematics

(4 Hours)(4 Hours)

PHYSICS CHAPTER 4

2

5.15.1 Angular parameters (1 hour)Angular parameters (1 hour)

At the end of this chapter, students should be At the end of this chapter, students should be

able to:able to:

�� DefineDefine angular displacement (θθθθ ), average angular velocity (ωωωωav),

instantaneous angular velocity (ωωωω),

average angular acceleration (ααααav), and

instantaneous angular acceleration (αααα).

�� UseUse and convertconvert the angular parameter

units.

Learning OutcomeLearning Outcome

PHYSICS CHAPTER 4

3

ANGULAR PARAMETERSANGULAR PARAMETERS

DisplacementDisplacement VelocityVelocity AccelerationAcceleration

Average AngularAverage Angular

Velocity, Velocity, ..avωAverage AngularAverage Angular

Acceleration, Acceleration, ..avα

Instantaneous Instantaneous

Angular Velocity,Angular Velocity,ω

Instantaneous Instantaneous

Angular Acceleration,Angular Acceleration,α

AngularAngular

Displacement, Displacement, θ∆

5.15.1 Angular Parameters Angular Parameters

PHYSICS CHAPTER 4

4

1. Definition

Angular displacement,θ is defined ___________

_________________________________________.*

or

as ______________________________________ ______________________________________

____________________________________________________________________________________

_________________________________________._________________________________________.*

(*as shown in Figure 5.1)

Figure 5.1Figure 5.1

5.1.1 Angular Displacement, 5.1.1 Angular Displacement, θ

PHYSICS CHAPTER 4

5

OR

where radianin nt)displaceme(angular angle :θlength arc :s

circle theof radius :r

2. Equation:

� Figure 5.1shows a point P on a

rotating compact disc (CD) moves

through an arc length s on a

circular path of radius r about a

fixed axis through point O. Thus,

Figure 5.1Figure 5.1

PHYSICS CHAPTER 4

6

1 rev2 rad360π==°

1803601 radππ

°°==

1 deg36 0180= =

o Sign convention of angular displacement

________________ – if the rotational motion is ____________________________.

________ ________ – if the rotational motion is ________________________.

3. SI Unit

o ________________

4. Unit Conversion

5.1.1 Angular Displacement, 5.1.1 Angular Displacement, (cont(cont……..)..)θ

PHYSICS CHAPTER 4

7

Example 1:

Convert the following angular displacement to

(a) in radian

(i) 60°

(b) in degrees

(i) π rad

Solution :

(a) (i)

(b) (i)

PHYSICS CHAPTER 4

8

5.1.2 Angular velocity5.1.2 Angular velocity

1. Average angular velocity1. Average angular velocity,, ωωωωav

i.i. DefinitionDefinition

Average angular velocity, ωωωωav is defined ___________

___________________________________________.

ii.ii. Equation :Equation :AV

ω

θ∆

where

radianin nt displacemeangular final :2θ

interval time :t∆

radianin nt displacemeangular initial :1θ

PHYSICS CHAPTER 4

9

2. Instantaneous angular velocity, 2. Instantaneous angular velocity, ωωωω

i.i. DefinitionDefinition

Instantaneous angular velocity, ωωωω is the _______________ _________________________________________________.

ii. Equation :

3. It is a3. It is a ______________________________________..

4. The unit of angular velocity is4. The unit of angular velocity is ______________________________________________

5. Other units are5. Other units are ________________________________________________________________________________

PHYSICS CHAPTER 4

10

6. Unit conversion :6. Unit conversion :

Note:

Every part of a rotating rigid body has the same angular

velocity.

PHYSICS CHAPTER 4

11

Example 2:Example 2:

Express (a) 100 revolutions per minute (rpm) in rad s-1

(b) 25 rad s-1 in revolutions per second

Solution :Solution :

(a) 100 rpm → rad s-1

PHYSICS CHAPTER 4

12

(b) 25 rad s-1 → rps

PHYSICS CHAPTER 4

13

Example 3:

If the angular the displacement, θ of rotating wheel is given by

determine,

(i) the average angular velocity at time t1 = 3.0 s and t2 = 5.0 s

(ii) instantaneous angular velocity at time t = 5.0 s

4 2

3t

+= πθ

PHYSICS CHAPTER 4

14

Solution :

(i) t1 = 3.0 s and t2 = 5.0 s

(ii) ω at t = 5.0 s

PHYSICS CHAPTER 4

15

5.1.3 Angular Acceleration5.1.3 Angular Acceleration

1. Average angular acceleration, ααααAV

i. Definition

Average angular acceleration, ααααAV is defined ________ ____________________________________________.

ii. Equation :

where

locityangular ve final :2ω

interval time :t∆

locityangular ve initial :1ω

PHYSICS CHAPTER 4

16

2. Instantaneous angular acceleration, 2. Instantaneous angular acceleration, αααα

i. Definition

Instantaneous angular acceleration is defined __________

____________________________________________.

ii. Equation :

5.1.3 Angular Acceleration5.1.3 Angular Acceleration (cont…..)

PHYSICS CHAPTER 4

17

3. It is a ___________________.

4. The unit of angular acceleration is __________________

Note:

�� If theIf the angular acceleration,angular acceleration,αα isis positive, positive, thenthen the angular the angular

velocity,velocity,ωω isis increasing.increasing.

� If the angular acceleration,α is negative, then the angular

velocity,ω is decreasing.

5.1.3 Angular Acceleration5.1.3 Angular Acceleration (cont…..)

PHYSICS CHAPTER 4

18

Example 4:

The instantaneous angular velocity, ω of the bicycle wheel is given by .

Calculate,

a. the instantaneous angular velocity at time, t = 2.0 s

b. the average angular acceleration between t1=2.0 s and t2=5.0 s

c. the instantaneous angular acceleration at time, t=3.0 s

23t2t6 +=ω

PHYSICS CHAPTER 4

19

Solution:Solution:

a. ω at time, t = 2.0 s

b. ωAV between t1= 2.0 s and t2= 5.0 s

3 2Instantaneous angular velocity, 6 2t tω = +

PHYSICS CHAPTER 4

20

Example 5:Example 5:

The initial velocity of a motorcycle is 2.2 rad s-1. After passing

over a hole, one of tires of motorcycle and velocity drops to

1.7 rad s-1. In a time interval of 5 minutes, calculate the average angular acceleration.

Solution:

PHYSICS CHAPTER 4

21

Exercise 5.1:Exercise 5.1:

1. The angular position θ of the blade of a fan is given by

Determine the angle θ, in radians and in degrees, at times

a. t1 = 1.0 s and

b. t2 = 3.0 s

Ans: 0.5 rad, 28.6º; 4.5 rad, 257.8º

2. If a disc 30 cm in diameter rolls 65 m along a straight line without

slipping, calculate

a. the number of revolutions would it makes in the process,

b. the angular displacement would be through by a speck of gum

on its rim.

ANS. : 69 rev; 138ANS. : 69 rev; 138ππππππππ radrad

( )-1 2 2.5 rad s tθ =

PHYSICS CHAPTER 4

22

Exercise 5.1:Exercise 5.1:

3. A particle moves along a circular path of radius 3.0 m with an

angular velocity 20 rad s-1.

Calculate,

a. the angular velocity in revolutions per second

b. the time for one revolution

c. displacement in one minute

Ans: 3.2 rev s-1; 0.31 s; 1200 rad

4. An engine requires 5 s to go from its idling speed of 600 rpm to 1200 rpm. What is its angular acceleration?

Ans: 12.6 rad s-1

PHYSICS CHAPTER 4

23

5.25.2 Relationship between linear & rotational Relationship between linear & rotational

motion (1 hour)motion (1 hour)

At the end of this chapter, students should be At the end of this chapter, students should be

able to:able to:

� Relate parameters in rotational motion

with their corresponding quantities in linear motion

Learning OutcomeLearning Outcome

22, , ,

t c

vs r v r a r a r

rθ ω α ω= = = = =

PHYSICS CHAPTER 4

24

Relationship between Relationship between

linear & rotational motionlinear & rotational motion

Relationship between Relationship between

linear velocity, linear velocity, vv and and

angular velocity, angular velocity, ωωωωωωωω

Relationship Relationship

between between

tangential tangential

acceleration, acceleration, aatt

and angular and angular

acceleration, acceleration, αααααααα

s rθ=

ωrv =

αrat =

22

c

va r

rω= =

PHYSICS CHAPTER 4

25

5.2 5.2 Relationship between linear & rotational Relationship between linear & rotational

motion motion

5.2.1 Relationship between linear velocity, v and

angular velocity, ωωωω

� When the rigid body is rotating , every particle of the rigid

body moves in a circle about rotation axis as shown in the

Figure 5.2.

vr

sθ

P

y

x

rFigure 5.2Figure 5.2

PHYSICS CHAPTER 4

26

5.2.1 Relationship between linear velocity, v and

angular velocity, ωωωω (cont…)

� Point P moves in a circle with tangential velocity, v where

the magnitude given by

� The ___________of the _____________________ always ____________________________.

PHYSICS CHAPTER 4

27

� Every particle on the rigid body has the _____________ ________ (magnitude of angular velocity) but the _____

______ is __ the _____ because the radius of the circle,ris _________________________________________.

� Each point or particle of a rotating rigid body moves in a circular motion with the axis of rotation as the centre.

The arc length s covered by the particle in time ∆t is

given by s = r θ

5.2.1 5.2.1 Relationship between linear velocity, Relationship between linear velocity, vv and and

angular velocity, angular velocity, ωωωωωωωω (cont(cont……))

Simulation 5.1Simulation 5.1Simulation 5.1

PHYSICS CHAPTER 4

28

� If the ___________ is ______________________ then the _______________ of a particle also_________ hence ___ component of _____________ are __________ as shown

in Figure 5.3.

5.2.2 Relationship between tangential acceleration, at

and angular acceleration, αααα

tar

cara

r

x

y

P

OFigure 5.3Figure 5.3

PHYSICS CHAPTER 4

29

� The components are _________________ and _________

________________ given by equations below.

5.2.25.2.2 Relationship between tangential acceleration, Relationship between tangential acceleration, aatt

and angular acceleration, and angular acceleration, αααααααα (cont(cont……..)..)

and

PHYSICS CHAPTER 4

30

� The ________________________________________ of

a particle in a rotating body is _____________________

given by

the magnitude,

5.2.25.2.2 Relationship between tangential acceleration, Relationship between tangential acceleration, atat

and angular acceleration, and angular acceleration, αααααααα (cont…..)

PHYSICS CHAPTER 4

31

Example 6:Example 6:

A particle moves along a circular path of radius 3.0 m with an angular velocity 20 rad s-1. Calculate,

a. angular acceleration at t = 3.0 s

b. distance traveled in one minute

c. the linear speed of the particle

d. the centripetal acceleration

Solution:

a.a.

PHYSICS CHAPTER 4

32

Solution:

b.b.

PHYSICS CHAPTER 4

33

Example 7:Example 7:

A wheel of radius 500 mm is spinning at a constant angular velocity of

Determine

a. angular acceleration at time, t = 2.0 minute

b. the velocity of the wheel at t = 3.0 s

c. the tangential linear acceleration and centripetal acceleration at t = 2.0 s

322

3

ttω = +

PHYSICS CHAPTER 4

34

Solution:Solution:

a.

b.

c.

PHYSICS CHAPTER 4

35

Example Example 8:8:

A steel cylinder 50 cm in radius is to be machine in a lathe. If the

desired linear velocity of the cylinder’s surface is to be 2.0 m s-2,

determine rpm should it rotate.

Solution:

PHYSICS CHAPTER 4

36

Exercise 5.2:Exercise 5.2:

1. A disk 8.00 cm in radius rotates at a constant rate of 1200 rev

min-1 about its central axis. Determine

a. its angular speed,

b. the tangential speed at a point 3.00 cm from its centre,

c. the radial acceleration of a point on the rim,

d. the total distance a point on the rim moves in 2.00 s.

ANS. : 126 ANS. : 126 radrad ss−−−−−−−−11; 3.77 m s; 3.77 m s−−−−−−−−11; 1.26 ; 1.26 ×××××××× 101033 m sm s−−−−−−−−22; 20.1 m; 20.1 m

2. A 0.35 m diameter grinding wheel rotates at 2500 rpm. Calculate

a. its angular velocity in rad s−1,

b. the linear speed and the radial acceleration of a point on the

edge of the grinding wheel.

ANS. : 262 ANS. : 262 radrad ss−−−−−−−−11; 46 m s; 46 m s−−−−−−−−11, 1.2 , 1.2 ×××××××× 101044 m sm s−−−−−−−−22

PHYSICS CHAPTER 4

37

Exercise 5.2:Exercise 5.2:

3. A rotating wheel required 3.00 s to rotate through 37.0 revolution. Its angular speed at the end of the 3.00 s interval is 98.0 rad s-1. Calculate the constant angular acceleration

of the wheel.

ANS. : ANS. : 13.6 13.6 radrad ss−−−−−−−−22

PHYSICS CHAPTER 4

38

5.35.3 Rotational motion with uniform Angular Rotational motion with uniform Angular

Acceleration (2 hour)Acceleration (2 hour)

At the end of this chapter, students should be able to:At the end of this chapter, students should be able to:

�� Use equationsUse equations for rotational motion with constant for rotational motion with constant

angular accelerationangular acceleration

�� Make analogyMake analogy with their corresponding quantities in with their corresponding quantities in

linear motionlinear motion

Learning OutcomeLearning Outcome

αtωω 0 +=

αθωω2

02 2+=

20 ttωθ α+=

2

1

PHYSICS CHAPTER 4

39

Rotational Rotational

Motion Equation Motion Equation AnalogyAnalogy

Linear Linear

MotionMotionAngular Angular

MotionMotion

Rotational motion with uniform Angular Rotational motion with uniform Angular

Acceleration Acceleration

tαωω += 0 αθωω 22

0

2+=

2

02

1tt αωθ +=

PHYSICS CHAPTER 4

40

5.3.15.3.1 Rotational motion with uniform angular Rotational motion with uniform angular

acceleration acceleration

� Table 5.1 shows the symbols used in linear and

rotational kinematics.

Table 5.1Table 5.1

QuantityRotational

motion

Linear

motion

PHYSICS CHAPTER 4

41where θ in radian.

Angular motionLinear motion

5.3.15.3.1 Rotational motion with uniform angular Rotational motion with uniform angular

acceleration acceleration (cont….)

� Table 5.2 shows the comparison of linear and rotational

motion with constant acceleration.

Table 5.2Table 5.2

PHYSICS CHAPTER 4

42

Example 9:

The tires of a car make 85 revolutions as the car reduces its speed

uniformly from 90.0 km h-1 to 60.0 km h-1. The tires have a diameter of 0.90 m. Determine the angular acceleration.

Solution:

PHYSICS CHAPTER 4

43

Example 10:Example 10:

An electric ceiling fan with blades 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of

0.250 revs-1. The angular acceleration is 0.900 revs-2.

a. Compute the angular velocity after 0.200 s.

b. Through how many revolutions has the blade turned in this

time interval?

c. What is the tangential velocity of a point on the tip of the

blade at t = 0.200 s?

d. What is the magnitude of the resultant acceleration of a

point on the tip of the blade at t = 0.200 s?

No. 9.26, pg. 354, University Physics with Modern Physics,11th edition, Young & Freedman.

PHYSICS CHAPTER 4

44

SolutionSolution:

PHYSICS CHAPTER 4

45

Example 11:Example 11:

The initial angular velocity of a body which rotates with uniform

angular accelerations is 11 rad s-1. After 2.0 s, its angular velocity is 19 rad s-1. Calculate the

(a) angular acceleration

(b) angular displacement after 2.0 s.

Solution:Solution:

(a)

PHYSICS CHAPTER 4

46

Example 12:Example 12:

A wheel rotates with uniform angular acceleration of 0.50 rad s-2.

(a) what is the time taken for its angular velocity to increase from 2.0 rad s-1 to 3.0 rad s-1.

(b) what is the angular displacement in this time?

Solution :Solution :

(a)

PHYSICS CHAPTER 4

47

Example 13:Example 13:

A fan is rotating at 1.2 revolutions per second. When switched off

it decelerates uniformly and stops in 5.0 s. Calculate

(a) angular decelerations

(b) number of revolutions made before it stops

Solution :Solution :

(a)

PHYSICS CHAPTER 4

48

Example 14:Example 14:

A small insect stands on a stationary circular disc. The disc starts

to rotate at a constant angular acceleration of 1.5 rad s-2. After 6.0 s the insect is thrown off the disc. Determine, at the moment the insect is thrown off,

(a) the number of revolutions the insects has turned through

(b) the angular velocity of the insect

Solution:Solution:

(a)

PHYSICS CHAPTER 4

49

Example 15:Example 15:

A spinning tire initially has an angular velocity of 30 rad s-1, 20 s

later its angular velocity is 45 rad s-1. If the angular acceleration is constant, determine

(a) the magnitude of the angular acceleration

(b) the angular displacement over 25 s

(c) the angular speed at 35 s

Solution:Solution:

(a)

PHYSICS CHAPTER 4

50

Example 16:Example 16:

(a) Through how many revolutions must the 55 cm diameter

tire of a truck turn as the truck travels 3.5 km?

(b) A CD turntable rotating at 9.0 rad s-1 slows uniformly to a stop in 8 rev. Calculate the angular acceleration in rad s-2.

Solution:Solution:

(a)

PHYSICS CHAPTER 4

51

Exercise 5.3:Exercise 5.3:

1. A wheel rotates with a constant angular acceleration of 3.50

rad s-2.

a. If the angular speed of the wheel is 2.00 rad s-1 at t = 0 s,

through what angular displacement does the wheel rotate

in 2.00 s.

b. Through how many revolutions has the wheel turned

during this time interval?

c. What is the angular speed of the wheel at t = 2.00 s?

pg. 297,Physics for scientists and engineers with modern physics, Serway &

Jewett,6th edition.

Ans. : 11.0 Ans. : 11.0 radrad, 1.75 rev, 9.00 , 1.75 rev, 9.00 radrad ss--11

PHYSICS CHAPTER 4

52

Exercise 5.3Exercise 5.3

2. A bicycle wheel is being tested at a repair shop. The angular

velocity of the wheel is 4.00 rad s-1 at time t = 0 , and its angular

acceleration is constant and equal to -1.20 rad s-2. A spoke OP

on the wheel coincides with the + x-axis at time t = 0 as shown in

figure below.

a. What is the wheel’s angular velocity

at t = 3.00 s?

b. What angle in degree does the spoke

OP make with the + x-axis at this

time?

Ans. : 0.40 Ans. : 0.40 radrad ss--11, 18, 18°°°°°°°°

x

y

PO

PHYSICS CHAPTER 4

53

Exercise 5.3

3. A phonograph turntable initially rotating at 3.5 rad s-1 makes three complete rotations before coming to a stop.

a. What is its angular acceleration?

b. How much time does it take to come to a stop?

pg. 121, College Matriculation Physics Semester I , Higher Learning Publication Sdn. Bhd.

Ans. : Ans. : --0.325 0.325 radrad ss--1, 1, --10.8 s10.8 s

4. A wheel rotates with a constant angular acceleration of 2.5 rad s-2. At time t = 0, the angular velocity of the wheel is 3.0 rad s-1. Determine,

a. the angle turned through by the wheel in the first 2 seconds,

b. angular velocity at t= 2.0 s

pg. 185, Physics Matriculation Study Guide Semester I , Cheong Foon Choong, Pearson Longman.

Ans. : 11 Ans. : 11 radrad, 8 , 8 radrad ss--1154

PHYSICS CHAPTER 4

THE END…

Next Chapter…CHAPTER 5 :

Oscillations