Physics

Session

Rotational Mechanics - 1

Session Opener

Earth rotates about its own axis in one day. Have you ever wondered what will happen to the length of a day if all the sea-water was to evaporate?

Session Objectives

Session Objective

1. Definitions of Angular Velocity & Acceleration(Instantaneous/Average)

2. Rotational Kinematics Equations

3. Relation between angular and Linear Quantities

Angular Displacement

It is the angle turned by a body or particle about a given axis of rotation

It is dimensionless Unit: Radians

Angular velocity

2 1

av2 1

–

t – t t

t 0

dlim

t dt

Instantaneous angular Velocity

Average angular Velocity

Unit : rad/s

Angular acceleration

2 1

av2 1

–

t – t t

t 0

dlim

t dt

Instantaneous angular acceleration:

Average angular acceleration:

Unit : rad/s2

LINEAR ANGULAR

Displacement

Velocity

Acceleration

12 xxx 12

dt

dxv

dt

d

dt

dva

dt

d

Linear and Angular Variables

Ox

y

v

r

S

rs

dt

drv

P

ra t

dt

d

ra 2r

2r

2t aaa

at

ar

a

Relationship Between Angular and Linear quantities

Ox

y

00

1

1

O

a

x

y

t=0x=x0

v0

t=tx=x1

v1

Rotational Kinematics Equations

21 0 0

1x x v t at

2 2

1 0 01

t t2

a is constant

1 0v v at

2 21 0 1 0v v 2a x x 2 2

1 0 1 02

is constant

1 0 t

Class Test

Class Exercise - 1

The length of the seconds hand of a clock is 10 cm. The speed of the tip of the hand is

(a) cm/ s (b) cm/ s4 2

(c) cm/ s (d) cm/ s3 8

Solution :

2 2v r r 10 cm/s

T 60 3Hence Answer is (c)

Class Exercise - 2

A body of mass 100 g revolves in a circle on a horizontal frictionless table attached by a cord 21 cm long to a pin set on the surface. If the body makes 2 complete revolutions per second, find the force exerted on it by the chord.

(a) 3.23 N (b) 3.40 N(c) 3.38 N (d) 3.32 N

Solution :

2 2

= 4 rad/s, r = 0.21 m, m = 0.1 kg.

F = mr = (0.1)(0.21)(4 ) = 3.32 N

Hence Answer is (d)

Class Exercise - 3

A motor car is traveling at 30 m/s on a circular road of radius 500 m. It is increasing in speed at a rate of 2 m/s2. What is its acceleration?

v = 30 m/s, r = 500 m, at = 2 m/s

2 2

2r

v (30)a 1.8 m/s

r 500

2 2 2r ta a a 2.7 m/s

Solution :

Class Exercise - 4

A constant torque is acting on a wheel. If starting from rest the wheel makes n rotations in t seconds, the angular acceleration is given by:

2 2

4 n 4 t(a) (b)

t n4 n

(c) 4 nt (d)t

Solution

0 = 0, = 2 n, = ?

20

1t t

2

Hence answer is (a)

21or 2 n t

2

24 n

ort

Class Exercise - 5

We know,

2d 9t 2t

–dt 20 3

29 5 10– 7.92 rad/s

20 3at t = 5s

Solution :

Hence answer is (b)

3 29t 2t60 6

A particle starts rotating from rest

according to the formula .

The angular velocity at the end of 5 seconds is

(a) 7.8 rad/s (b) 7.92 rad/s

(c) 7.5 rad/s (d) 9.72 rad/s

3 29t 2t60 6

Class Exercise - 6

In the previous question the angular acceleration of the particle at the end of 5s will be

(a) 3.38 rad/s2 (b) 8.38 rad/s2

(c) 8.33 rad/s2 (d) 3.83 rad/s2

2d 18t 2

– 3.83 rad/sdt 20 3

Solution :

Hence answer is (d)

Class Exercise - 7

A flywheel rotating at the rate of 120 rpm slows down to rest at a constant rate of 2 rad/s2. How many rotations does it make in the process?

2(a) 2 rev (b) 4 rev

(c) 4 2 rev (d) 2 rev

Solution

Hence answer is (d)

1

22

= 120 rpm = 4 rad/s,

= 0, = -2 rad/s

2 22 1 = + 2(- )

2 21 216

or 42 4

24Number of rotations 2 .

2

Class Exercise - 8A flywheel is rotated at a speed of 60 rad/s. Because of friction on the axle, it comes to rest in 5 minutes. The angular retardation is

(a) 0.3 rad/s2 (b) 2 rad/s2

(c) 3 rad/s2 (d) 0.2 rad/s2

0 = 60 rad/s, = 0, t = 5 min = 300 s.

0 = + (- )t

0 260

or 0.2 rad/st 300

Solution :

Hence answer is (d)

Class Exercise - 9A wheel rotating at a speed of 600 rpm about its axis is brought to rest by applying a constant torque for 10 seconds. The angular velocity after 5 seconds is(a) rad/s (b) 2 rad/s

(c) 5 rad/s (d) 3 rad/s

Solution

Hence answer is (a)

0 = 60 rpm = 1 rps = 2 rad/s

= 0, t = 10 s

2

02

t or rad/s10 5

Angular velocity after 5 s is given by

0 (– )5

2 – 5 rad/s5

Class Exercise - 10

A wheel is subjected to uniform angular acceleration about its axis. Initially its angular velocity is zero. In the first 2 s, it rotates through an angle 1, in the next 2s it rotates through an angle 2. The

ratio of is2

1(a) 1 (b) 2

(c) 3 (d) 5

Solution

21

1(2) 2

2

Hence answer is (c)

21 2

1(4) 8

2

2or 8 – 2 6

2

1

63

2

Thank you