22. Rotational Mechanics-4

7/31/2019 22. Rotational Mechanics-4

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Physics


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Session

Rotational Mechanics - 4


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Session Objectives


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Session Objective

1. Angular Momentum of a Particle

2. Conservation of angular momentum

3. Angular Impulse


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Angular Momentum

Angular momentum of a particle

Angular Momentum of P L

L r p (p mv)

mr ( r)

2

L is an axial vector units : Kg m /s

2

L mvr sin

L I ( I mr )

Particle P rotates around Point O

Or

P(m)

v


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Angular Momentum for a Systemof Particles

For a system of particles (i = 1 to n)

n

i i

i 1

L r p I

2

i i( is a constant. So I m r )

Angular momentum depends on

the position of the axis


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Angular Momentum for a RigidBody

For a rigid body

L I

( I : moment of inertia around

the axis of rotation)

X

Y

O m

r

Z


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L l

dL dl l

dt dt

d

dt

ext extdL ldt

ext i i f f If 0 l l

ext

dLIf, 0 then, 0

dt

L cons tant

Conservation of Angular Momentum


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Conservation of Angular Momentum

If no external torque acts on a body

so the total angular momentum

of the body (or system of bodies) remainsa constant , and the vector sum of all torquesadd up to zero.

dL

0d t


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Equilibrium of a Rigid Body

Two conditions are necessary.

(i) Total force acting on the bodymust add up to zero (equilibrium oflinear motion)

(ii) Total torque acting on the bodymust add up to zero (equilibrium of

rotational motion)


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LINEAR ANGULAR

Mass

Momentum

Newtons

Second Law

Work

KineticEnergy

Power(constant force)

FdxW dW

2

mv2

1

K

2I

2

1K

Comparison

ILP=Mv

IF=ma

m I

PP=Fv


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Class Test


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Class Exercise - 1

A planet revolves round the sun as

shown. The KE is greatest at

A

B

C

D

(a)A (b) B

(c) C (d) D


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Solution

Kinetic energy at any point:

2 2

2

L LK

2I 2mr

L is a constant for the system (as motion of sunis negligible). L is constant for planet. So K is

maximum for smallest r which is A.

2

1K

r

Hence, answer is (a).


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Class Exercise - 2

A horizontal platform with a mass of

100 kg rotates at 10 rpm about avertical axis passing through its centre.A man weighing 60 kg is standing onthe edge. With what velocity will theplatform begin to rotate if the man

moves from the edge to the centre?

(a)22 rpm (b) 11 rpm

(c) 44 rpm (d) 66 rpm


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Solution

Moment of inertia of platform:

2P 11

m r I2

Moment of inertia of man at edge:

2m 2m r I

Moment of inertia of man at centre: O ( r = 0)


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Solution contd..

L is conserved:

1 2 1 1 2

2

2

2

I I I

1 1

100 60 10 1002 2

1100 50

22 rpm



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Class Exercise - 3

A mass moves about the Y-axis

with acceleration ay = (by2 c); band c are constants. The value of yfor which the angular momentumis zero, is

3c c(a) (b)b b

3c 3b(c) (d)

b c


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Solution

y 2

y

dvAcceleration a by c

dt

y 2y

vy y

2y y

0 0

32

y

dvv by c

dy

v dv by c dy

by

v cy 23 L (= mvyr) is zero when vy = 0

3

by 3ccy y

3 b

Hence, the answer is (c).


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Class Exercise - 4

A thin circular ring of mass M and radius

R is rotating about its axis with a constantangular velocity . Two objects each ofmass m are attached gently to the ring.The ring now rotates with an angularvelocity.

M 2mM(a) (b)

M m M 2m

M 2mM(c) (d)

M 2m M


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Solution

2 2I (ring) = MR , L = I = MR

when two masses are attached,M.I. becomes

2 2 2 2I' MR mR mR (M 2m)R

L I I' '

I M'

I' M 2m

Hence, answer is (c).


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Class Exercise - 5

A mass M is moving with a constant

velocity parallel to the X-axis. Itsangular momentum with respect tothe origin

(a) is zero (b) remains constant

(c) goes on increasing (d) goes on decreasing


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Solution

Let the motion of P be in xy plane.

z = 0

x

y

P

O

b

Vx

Then y = b is constant.

Velocity Vx is along x-axis (constant) (Vy, Vz = 0)

Z y x

x

L kL k xV yV m

k mbV (Constant)

Hence, answer is (b).


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Class Exercise - 6

If the rotational kinetic energy and

translation kinetic energy of arolling body are same, the body is

(a)disc (b) sphere

(c) cylinder (d) ring


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Solution

KE (translatory) = mv21

2

2

2

2

1 1 vKE(rotatary) I I

2 2 R

I always has the form kmR2

, where k is a fraction or unity.

2

2 2

2

1 v 1KE (rotatory) kmR kmv

2 2R

KE (rotatory) = KE (translatory) if k = 1 or I = mR2.This is true for a ring.

Hence, answer is (d).


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Class Exercise - 7

Two gear wheels, A and B, whose radii

are in the ratio RA : RB = 1 : 2, areattached to each other by an endlessbelt. They are mounted with their axesparallel to each other. The system oftwo wheels is set into rotation. It is

seen that both have the same angularmomentum. What is the ratio of theirmoments of inertia? (Belts do not slip)

(a)1 : 2 (b) 1 : 1

(c) 1 : 4 (d)1: 2


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Solution

Linear speed v of both wheels is the same.

A A B Bv R R

A BA B

B A

R2 2

R

A A B BL I I

A B

B A

I 1

I 2



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Class Exercise - 8

Consider the previous problem. If the

mass ratio mA : mB = 1 : 4, what isthe ratio of their rotational kineticenergies?

(a)4 : 1 (b) 2 : 1

(c) 1 : 2 (d) 1 : 4


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Solution

IA : IB = mARA2 : mBRB

2

= mARA2 : 4mA(2RA)

2

= 1 : 16

A B= 2

22 2 A

A A B B A1 1 1

I , I 16I2 2 2 2

2A A1 16

I2 4

A B(KE) 4 KE

Hence, answer is (d).


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Class Exercise - 9

Two masses mA and mB are attached to

each other by a rigid, mass less rod oflength 2r.They are set to rotation aboutthe centre of the rod with an angular speed. Then their angular momentum is

2 r

r r

z

xM

A MB

2

A B

1

(a) m m r along the x axis2

2A B(b) m m r along the y axis

2A B1

(c) m m r along the z axis

2 2A B(d) m m r in the x y plane


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Solution

2 2

A A B B

1 1

I m r , I m r2 2

2A B1

I m m r2

2A B1

L m m r2

As the rotation is in the xy plane, is along the z-axis.L

Hence, answer is (c).


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Thank you

Documents

22. Rotational Mechanics-4