phy 400 - chapter 6 - rotational motion

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PHY400

Physics for Non-Majors

Assoc. Prof. Dr. Ahmad Taufek Abdul Rahman PhD (Medical Physics), University of Surrey, UK

M.Sc. (Radiation Health Physics), UTM

B.Sc. Hons. (Physics & Math), UTM

ahmadtaufek@ns.uitm.edu.my

ahmadtaufek.ns@gmail.com

Chapter 6

ROTATIONAL MOTION

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6. Rotational Motion

– Angular Quantities

– Relationship Linear-Circular Motion

– Rotational Motion Equation

– Centripetal Force

Chapter 6

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Angular Quantities

The motion a rigid body can be analysed as the translational

motion of its centre of mass plus rotational motion about its centre

of mass.

For the ideal rotational motion of an object about a fixed axis, it

mean that all point / particle of the system move in circles

Chapter 6

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Angular Quantities Consider to the diagram, when system rotate, P is moving in circles

at uniform distance R from the axis of rotation O. The point moves

through an angle when its travel the distance l measured along

the circumference of its circular path.

is stated in radian, 1 radian is defined as the angle by a rotation

with l (length) is equal to R (radius)

= angles

l = length of circumference

R = radius

Chapter 6

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Angular Quantities

Angular displacement,

Chapter 6

, stated in radian

Derive that 360 = 2 radian

R

l

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Angular Quantities Consider a bicycle wheel rotate from one initial position, let say 1,

to some final position 2, therefore, the different between this two

positions stated as angular displacement (),

The angular displacement measured in radian (rad)

Chapter 6

= 2 - 1

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Angular Quantities Angular Velocity,

Consider a bicycle wheel rotate from one initial position, 1, to

some final position 2, thus the time rate of change of angular

displacement is defined as average angular velocity, .

The instantaneous angular velocity is the limit of this ratio as t

approaches zero (t 0)

The angular velocity has units of radian per second (rad/s)

Chapter 6

= / t

dt

d

t

0lim

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Angular Quantities Angular Acceleration,

Consider a bicycle wheel rotate from one initial position, 1, to

some final position 2, thus the time rate of change of angular

velocity is defined as average angular acceleration, .

The instantaneous angular acceleration is the limit of this ratio as

t approaches zero (t 0)

The angular acceleration has units of radian per second squared

(rad/s2)

Chapter 6

= / t

dt

d

t

0lim

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Angular Quantities

Chapter 6

dt

d

t

0lim

dt

d

t

0lim

= 2 - 1

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Angular Quantities

The frequency is the number of complete revolutions per

second:

Frequencies are measured in hertz:

The period is the time one revolution takes:

Chapter 6

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Linear vs. Angular Motion

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

parameters in rotational motion with their corresponding quantities

in linear motion. Write and use;

Chapter 6

r

vrararvrθs ct

22 ; ; ;

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Linear vs. Angular Motion

Consider a point P located a distance R from the axis of rotation, if

the body rotates with angular velocity , any point will have a linear

velocity whose direction is tangent to its circular path

Chapter 6

Rvdt

dRv

RdlR

l

dt

dlv

2

1

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Linear vs. Angular Motion

If the angular velocity of a rotating object changes, the object as a

whole and each point/particle in it has an angular acceleration.

Each point also has a linear acceleration whose direction is

tangent to that point’s circular path.

Chapter 6

Radt

dRa

RdRv

dt

dva

tantan

tan

2

1

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Linear vs. Angular Motion

The total linear acceleration is equal to sum of tangential

acceleration (atan) and centripetal acceleration (aR)

Chapter 6

Ra

RR

R

R

va

aaa

R

R

R

2

2

22

tan

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Linear vs. Angular Motion

Chapter 6

Linear Quantities

s = Displacement

v = velocity

atan = acceleration

Angular Quantities

= Displacement

= velocity

= acceleration

Linear Quantities Vs. Angular Quantities

s = R

v = R

atan = R

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Rotational Motion Equation

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

use equations for rotational motion with constant angular

acceleration;

Chapter 6

t 2

1

αtωω 0

2

02

1ttωθ

αθωω 22

0

2

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Example 1

a) Write out the definitions for all of the rotational quantities.

b) Write the equations relating the angular quantities to the

linear quantities.

c) Write down the equations for rotary motion with constant

acceleration.

d) Write the rotational versions of:

i. F = ma

ii. W = Fd

iii. E = ½ mv2

Chapter 6

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Example 2

The moon orbits the earth every 29 days. Find its angular velocity.

2.5 x 10-6 rad/s

Chapter 6

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Example 3

a) Find the radial acceleration of an 82 kg person at the earth’s

equator.

b) Calculate the centripetal force.

c) Calculate the normal force exerted by the earth

0.0337 m/s2 2.76 N 801 N

Chapter 6

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Example 4

a) Determine the tangential velocity of the earth about the sun.

(REarth’s Orbit=1.5 x 108 km)

b) The tangential velocity at the equator of a large sphere of radius

35 cm increases from 2.2 m/s to 8.4 m/s in 18 rotations. Find the

acceleration.

c) A tangential force of 22.5 N applied to a 4.5 kg sphere results in

an angular acceleration of 78.2 rad/s2. Find the radius of the

sphere.

2.99 x 104 m/s

8. 2.37 rad/s2

9. 16.0 cm

Chapter 6

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Example 5

A motorbike moving at a constant speed 20.0 m s1 in a circular track

of radius 25.0 m. Calculate

a. the centripetal acceleration of the motorbike,

b. the time taken for the motorbike to complete one revolution.

Chapter 6

2s m 16.0 ca s 7.85T(towards the centre of the circular track)

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Example 5

A boy whirls a marble in a horizontal circle of radius 2.00 m and at

height 1.65 m above the ground. The string breaks and the marble

flies off horizontally and strikes the ground after traveling a

horizontal distance of 13.0 m. Calculate

a. the speed of the marble in the circular path,

b. the centripetal acceleration of the marble while in the circular

motion.

(Given g = 9.81 m s-2)

Chapter 6

1s m 22.4 u2s m 512 ca (towards the centre of the horizontal circle)

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Centripetal Force From Newton’s second law of motion, a force must be associated

with the centripetal acceleration. This force is known as the

centripetal force and is given by

Chapter 6

amFF nett

cc amF

mvmrr

mvF 2

2

c

caa

vrr

va 2

2

c

where cFF

and

and

force lcentripeta :cFwhere

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Centripetal Force • The centripetal force is defined as a force acting on a body

causing it to move in a circular path of magnitude

and its always directed towards the centre of the circular path.

• Its direction is in the same direction of the centripetal

acceleration as shown in Figure.

Chapter 6

r

mvF

2

c

ca

cF

cF

cF

ca

ca

v

v

v

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Centripetal Force • If the centripetal force suddenly stops to act on a body in the

circular motion, the body flies off in a straight line with the

constant tangential (linear) speed as show in Figure.

Chapter 6

cF

ca v

cF

cF

ca

ca

v

v

v

v

0Fc

0Fc

0ac

0ac

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Centripetal Force • Note :

– In uniform circular motion, the nett force on the system is

centripetal force.

– The work done by the centripetal force is zero but the

kinetic energy of the body is not zero and given by

Chapter 6

222 mr2

1mv

2

1K

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Example 6

Figure shows a conical pendulum with a bob of mass 80.0 kg on a

10.0 m long string making an angle of 5.00 to the vertical.

a) Sketch a free body diagram of the bob.

b) Determine

i. the tension in the string,

ii. the speed and the period of the bob,

iii. the radial acceleration of the bob.

(Given g =9.81 m s2)

Chapter 6

N 788T 1s m 0.865 v2s m 0.859 ras 6.33T

(towards the centre of the horizontal circle)

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Example 7

A car of mass 2000 kg rounds a circular turn of radius 20 m. The road

is flat and the coefficient of friction between tires and the road is 0.70.

a) Sketch a free body diagram of the car.

b) Determine the maximum car’s speed without skidding.

(Given g = 9.81 m s-2)

Chapter 6

1s m 11.7 v

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Example 8

A ball of mass 150 g is attached to one end of a string 1.10 m long.

The ball makes 2.00 revolution per second in a horizontal circle.

a) Sketch the free body diagram for the ball.

b) Determine

i. the centripetal acceleration of the ball,

ii. the magnitude of the tension in the string.

Chapter 6

r

2s m 173 ca (towards the centre of the horizontal circle) N 26.0T

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Thank You & All the Best

“The actions which the Messenger of Allah, may Allah bless him and grant him peace, loved most were those which were done most constantly.”

Narrated by A’isha (ra) in Al-Muwatta, vol 9, number 92b