Physics

Session

Rotational Mechanics - 3

Session Objectives

Session Objective

1. Torque Definition

2. Torque and Angular Acceleration Relationship

3. Kinetic Energy in Rotational Motion

4. Power Delivered in Rotational Motion

5. Work Done in Rotational Motion

F

rO

O

d

Moment arm

Line of action

FsinF

cosF

r

r

Torque Definition

r F

r F sin( )

F d

X

Y

O m

r

Remember F=ma?

What m is to force, I is to torque.Z

What v is to P, is to L

Remember P=mv?Where P is Linear momentum.

Torque and Angular Acceleration Relationship

z zl

z zL l

Since is constant

I is Moment of Inertia

Po

ri

Vi

mi

y

x

Kinetic Energy in Rotational Motion

2i i i

1K mv2

2 2 2B i i i i i

1 1K K mv m r2 2

2 2B i i

1K m r2

2B

1K l2

2i il m r

Power Delivered in Rotational MotionThe rate of increase of kinetic energy equals the rate at which work is done on it, i.e. the power delivered by the torque

2dW dK d 1 dP I Idt dt dt 2 dt

I

Work Done in Rotational Motion

dW P.dt dt .d

Work done in an infinitesimal angular displacement d

2

1

W .d

Thus, the work done during an angular displacement 1 2to is

Class Test

Class Exercise - 1One end of a ladder is resting on a wall and the other end on the ground. Assume the weight of the ladder to be mg, the length of the ladder and the angle of inclination to the horizontal is . Find out the total torque about the lowermost point of the ladder.

SolutionTaking moment about A

mg

B N

R

A

mg cos N sin2

Where N and R are the reactions of the wall andthe floor respectively.

Class Exercise - 2Find out the total torque acting about point A in the diagram shown below.

Length = lm ass = M

F

A

Solution

90 –

lA

= r × F = Force × Perpendicular distance = F × cos = F cos

Class Exercise - 3

O20

F F

40 60 80FF

Four equal and parallel forces are acting on a rod as shown in figure at distances 20 cm, 40 cm, 60 cm and 80 cm respectively from one end of the rod under the influence of these forces the rod(a) Is at rest(b) Experiences a torque(c) Experience a linear motion(d) Experience of torque and also linear motion

SolutionAs net force is zero it will notperform linear motion.

Now taking torque about O,

netnet

= -20F + 40F - 60F + 80F = 40F

Hence, it experiences a torque.Therefore the answer is (b).

Class Exercise - 4The moment of inertia of a wheel about the axis of rotation is 3.1 mks units. Its kinetic energy will be 600 J if the period of rotation is

(a) 0.05 s (b) 0.314 s

(c) 3.18 s (d) 20 s

Solution

21KE I

2

2 12003.1

12003.1

2T

2

T

T = 0.314 s

Hence, answer is (b).

Class Exercise - 5Four masses are fixed on a massless rod as shown in the following figure. The moment of inertia about the axis P is

(a) 2 kg m2 (b) 1 kg m2

(c) 0.5 kg m2 (d) 0.3 kg m2

2 kg 2 kg5 kg 5 kg

0.2 m 0.2 m 0.2 m 0.2 m

P

Solution

2I mr

2 2I 5 0.2 2 0.4 2

I = 0.3 kg-m2

Hence, answer is (b).

Class Exercise - 6Two racing cars of masses m1 and m2 are moving in circles of radii r1 and r2 respectively. Their speeds are such that they make a complete circle in the same length of time t. The ratio of the angular speed of the first to the second car is

(a) m1 : m2 (b) r1 : r2

(c) 1 : 1 (d) m1r1 = m2r2

Solution

1 2

1 2

2 r 2 rv v

1 1

2 2

v rv r

v

Butr

1

2

11

Alternative solution

2

T is same.T

1

2

1Hence, will be same

1

Hence, answer is (c).

Class Exercise - 7A meter stick is held vertically with one end on the floor and is allowed to fall. The speed of the other end when it hits the floor (assume that the other end at the floor does not slip).

(a) 3.2 m/s (b) 5.4 m/s

(c) 7.6 m/s (d) 9.2 m/s

Solution

Conservation of energy:

A

B

2

A1

mg I2 2

2 3g

3g v 3g

= 1 m v 30 v = 5.4 m/s

Hence, answer is (b).

Class Exercise - 8An inclined plane makes an angle of 30o with the horizontal. A ring rolling down the inclined plane from rest without slipping has a linear acceleration equal to

2g g(a) (b)3 2g g(c) (d)3 4

Solution

30°

2

gsina

I1

MR

2

2

gsin30 ga sin30

2MR1

MR

g 1

a2 2

g

a4 Hence, answer is (d).

Thank you