2018-10-01_Physics_For_You

Relation between Angular Momentum and

Moment of Inertia
Angular momentum, L = IZ
e angular acceleration D of a rigid body rotating about
a xed axis is given by ID = W. If the external torque W is
zero, the component of angular momentum about the
xed axis IZ of such a rotating body is constant.

A ROLLING RIGID BODY

Kinetic energy of rotational motion, KIR=

1

2

ω^2.

Kinetic energy of a rolling body = translational kinetic
energy (KT) + rotational kinetic energy (KR).

=+=+

















1

2

1

2

1

2

(^2221)
2
Mv IMv 2

K

R

ω

When a body rolls down an inclined plane of inclination
q without slipping its velocity at the bottom of incline is

given by v gh
K
R

=

+

2

1

2

2

where h is the height of the incline.
When a body rolls down on an inclined plane without
slipping, its acceleration down the inclined plane is

given by a

g

K

R

=

+

sin

.

θ

1

2

2

When a body rolls down on an inclined plane without
slipping, time taken by the body to reach the bottom is

given by t

l K

R

g

=

+









21

2

2

sinθ

where l is the length of the inclined plane.

ANALOGY BETWEEN TRANSLATIONAL MOTION AND

ROTATIONAL MOTION

Translational

motion

Rotational motion

about a xed axis

1. Displacement x Angular displacement q


2. Velocity v = dx/dt Angular velocity
   Z = dq/dt


3. Acceleration
   a = dv/dt

Angular acceleration

D = dZ/dt

4. Mass M Moment of inertia I


5. Force F = Ma Torque W = ID


6. Wo r k dW = Fds Wo r k dW = Wdq


7. Kinetic energy of a
   translational motion
   KT = Mv^2 /

Kinetic energy of a

rotational motion

KR = IZ^2 /

8. Power P = Fv Power P = WZ


9. Linear momentum
   p = Mv

Angular

momentum L = IZ

10. Equations of
    translational motion
    (i) v = u + at
    (ii) su=+tat

1

2

2

(iii) v^2 – u^2 = 2as

(iv)su

a

nth=+ n−

2

() 21

Equations of

rotational motion

Z = Z 0 + Dt

θω=+ 0 α^2

1

2

tt

Z^2 – Z 02 = 2Dq

θω

α

nth=+ 0 2 ()^21 n−

THOUGHT PROVOKING POINTS

• Moment of inertia of a body has dierent values
  in dierent directions and as such it is not a scalar.
  Futher, it is not a vector either as its value about a
  given axis remains the same whether the direction
  of rotation is clockwise or anticlockwise, i.e.,
  direction of rotation need not be specied. In fact,
  it is tensor quanity.


• Rolling motion is possible only if a frictional force
  is present between the rolling body and incline to
  produce a net torque about the centre of mass.


• e angular velocity of rotating body is the same
  relative to any point on it.


• If sense of angular velocity and angular acceleration
  is same, the rigid body is speeding up and if they
  have opposite sence, the body is slowing down.


• If an object is uncostrained (i.e., not an pivot)a
  couple causes the object to rotate about its centre
  of mass.


• e gravitational torque on any extended object
  is equivalent to the torque of a single force,
  gravitational force, acting at the objects centre of
  mass. We can treat the object as if all its mass were
  concentrated at the centre of mass.


• Rotational kinetic energy is maximum for ring and
  minimum for solid and moving with same speed.