2018-10-01_Physics_For_You

∫dm=M()massofthebody

r

M

cm = ∫rdm

1

.

Note: If an object has symmetric uniform mass

distribution about x axis then y coordinate of COM is

zero and vice-versa

Centre of Mass of a Uniform Rod

Suppose a rod of mass M and length L is lying along the

x-axis with its one end at x = 0 and the other at x = L.

Mass per unit length of the rod = M/L

Hence, dm, (the mass of the element dx situated at

x = x is)=

M

L

dx

e coordinates of the element PQ are (x, 0, 0).
erefore, x-coordinate of COM of the rod will be

x

xdm

dm

x

M

L

dx

ML

xdx

L

L L

L

cm ==









==

∫

∫

∫

∫

0 0

0

1

2

()

e y-coordinate of COM is y

ydm

cm dm

==∫

∫

0

Similarly, zcm = 0

i.e., the coordinates of COM of the rod are

L

2

,, 00.







^

Or it lies at the centre of the rod.

RIGID BODY

A rigid body is one whose geometric shape and size

remains unchanged under the action of any external

force. When a rigid body performs rotational motion,

the particles of the body move in circles. e centres

of these circles lie on a straight line called the axis of

rotation, which is xed and perpendicular to the planes

of circle.

Torque and Angular Momentum

Torque is the turning eect of a force. If a force acting

on a object has a tendency to rotate the body about an

axis, the force is said to exert a torque on the body. It is

a vector quantity. In vector form,

Torque, τ=×



rF
In magnitude, W = r F sinq.
Here q is the angle between 


rFand.
Torque has the same dimensions as that of work i.e.
[ML^2 T–2]. But work is a scalar quantity whereas torque
is a vector quantity.

By convention, anticlockwise moments are taken as

positive and clockwise moments are taken as negative.

Special Cases

• 
  

(the rod will not rotate)

Ifqt= 0°, then = 0

O P

axis of rotation

r F

• (^) O P
If = 180°, then = 0
(the rod will not rotate)
qt
axis of rotationF
r

• 
  

Ifq= 90°, thent=rF(maximum torque)

O P

axis of rotation F q= 90°

r

Note : Same force acting at the same point can produce

either anticlockwise or clockwise torque depending

upon the location of the axis of rotation as shown in

the gure.

A B

Axis of rotation

F

Anticlockwise torque

C

Axis of rotation

F

Clockwise torque

C

A B

Work Done by Torque

Work done, W = torque u angular displacement

= W × 'q

Power, P

dW

dt

d

dt

===τ

θ

τω

Angular momentum of a particle about a given axis is

the moment of linear momentum of the particle about

that axis. It is denoted by symbol



L.

Angular momentum

  

Lr=×p

In magnitude, L = rp sinq

where q is the angle between rpand.

Angular momentum is a vector quantity. Its SI unit is

kg m^2 s–1. Its dimensional formula is [ML^2 T–1].

Relationship between Torque and Angular

Momentum

Rate of change of angular momentum of a body is equal

to the external torque acting upon the body.





τext=

dL

dt