CENTRE OF MASS, RIGID BODIES, ROTATIONAL MOTION

The centre of mass of n particles of masses m1, m2, ….mn, located at points specified

by position vectors r1, r2, …….rn is at a point of position of vector R where

R =

For a continuous body, the centre of mass is denoted by

R =

Where r1 is the position vector of the point around which the element of mass dm1 is

situated. Also, the integration is to be carried so as to cover the whole body.

When a particle/particles or some portion of a continuous body is removed and we have

to find the centre of mass of the left over portion, we can do that by using the above

formulae but by attaching a negative sign to the masses of the particle/particles or the

portion of the body removed.

The location of the centres of mass of some regular uniform bodies is shown in the

figures below :

The torque | | of a force | F | about a given centre of rotation is the product of

(a) The force and its lever-arm, i.e., the perpendicular distance of its line of action

from the centre of rotation.

(b) The angular component of the force | F | and the radial distance r of its point of

application from the centre of rotation.

Torque = r × F

The power P associated with a torque is defined by

P =

The angular momentum L of a rotating particle about a given centre of rotation is viewed

as the ‘moment of its momentum’ and is defined by

L = r × p

where p is the momentum of the particle (p = mv).

The torque and the angular momentum L are related as

=

The angular momentum of a particle also equals the product of double its mass and its

areal velocity.

When the external torque acts on a given system of particles, its total angular

momentum remains constant, i.e., it is conserved.

The centripetal force F acting on a particle moving with a speed v in a circle of radius r

is given by

F = mv2/r = mr

When a vehicle moves on a curved horizontal road, its maximum safe speed is given by

vmax =

Where , is the coefficient of static friction between the vehicle wheel and the road.

For a curved road of radius r banked at an angle �, the maximum safe velocity v0 is

given by

v0 =

where is the coefficient of kinetic friction between the vehicle wheels and the road.

For a ‘banked road’ when no friction comes into play, we have the maximum safe

speed, u0, given by

u0 =

The safe speed v of a vehicle moving on a ‘banked road’ of radius r lies between the

limits u0 and v0, i.e.,

vsafe

When a particle moves in a vertical circle, the tension T in the string is given by

T = + mg cos�

Where v is the speed of the particle at the point considered and � is the angle made by

the line, joining this point to the centre, with the vertical.

At the lowest point, T is the maximum and is given by

T1 = + mg

At the highest point, T is the minimum and is given by

Th = - mg

The minimum velocity with which a particle must start from the lowest point to complete

a vertical circle is given by

vmin =

The moment of inertia, I, about a given axis of rotation, of a system of n particles of

masses m1, m2 , ….mn located at point distant r1 , r2 , …..rn from the axis of rotation is

given by

I =

Where K is radius of gyration.

Kinetic energy of Rotation. If I is the moment of inertia of a body rotating with angular

velocity then its rotational kinetic energy = I .

If a body slips, it has only translational kinetic energy = Mv2cm, vcm being velocity of

centre of mass.

If a body rolls, it has simultaneous rotational and translational motion; then its total

kinetic energy

= I + V2cm

Theorems of Moment of Inertia. These theorems are very important because they

help in determining the moment of inertia about any axis if moment of inertia about

some axis is known.

1. Theorem of parallel axes. This theorem states that the moment of inertia of a

body about any axis is equal to its moment of inertia about a parallel axis through

its centre of mass plus the product of the mass of the body and square of

perpendicular distance between the two axes, i.e.

I = IG + Mr2

Where IG is moment of inertia about an axis passing through centre of gravity, M is

total mass of body and I is the moment of inertia about a parallel axis at a distance r

from that passing through centre of gravity.

2. Theorem of perpendicular axes. This theorem states that the sum of the

moments of inertia of a plane lamina about any two mutually perpendicular axes

in its plane is equal to its moment of inertia about an axis perpendicular to the

plane of the lamina and passing through the point of intersection of the first two

axes i.e.,

Iz = Ix + Iy

Where x, y axes are usual mutually perpendicular axes.

Let a body of mass M and radius R roll down (without slipping) an inclined plane

of slope �. As the body rolls down, its potential energy is converted to kinetic

energy of translation plus the kinetic energy of rotation i.e.

Mgh = I + M , …(1)

Where h is the height through which the body falls. If k is radius of gyration.

v2 =

or a = =

The following table summarizes the rotational motion analogous of the basic concepts

for linear or translator motion.

Linear motion Rotational motion

Point Mass A Rigid Body

Position, x Angle, �

Displacement Velocity Angular velocity

(v = ) ( = )

Acceleration Angular Acceleration

(a = ) ( = )

Mass, m Moment of inertia (I)

(about a given axis)

Momentum (p = mv) Angular momentum

(L = I )

Force, F Torque

F = =

Kinetic energy = mv2 Kinetic energy = I

Table of Values on Moment of Inertia

Body Axis Moment of

Inertia

(1) (2) (3)

1. Thin uniform rod (i) Through its centre and perpendicular

of length L. to its length. Through one end and

perpendicular to its length.

2. Thin rectangular (i) Through its centre and parallel to one

lamina of sides side (a or b)

a and b

(ii) About one side or

(iii) Through its centre and perpendicular M( )

to its plane.

(iv) Through midpoint of one side (a or b) M( + )

and perpendicular to its plane.

3. Thick uniform Through its midpoint and perpendicular M( )

rectangular bar to its length.

of length, breadth

and thickness a, b

and c resp.

4. Circular ring of (i) Through its centre and perpendicular

radius R. to its plane.

(ii) About a Diameter.

5. Circular lamina or (i) Through its centre and perpendicular MR2

disc or disc of radius to its plane

R. (ii) About a diameter

6. Hollow cylinder of Axis of the cylinder MR2

radius R.

7. Solid cylinder of Axis of the cylinder MR2

radius R.

8. Solid sphere of About a diameter

radius R.

When a body rolls without slipping on an inclined plane, its linear and angular

acceleration are given by

a =

and a =

The maximum allowed inclination of the plane so that a body may roll down it without

slipping is given by = tan-1 where is the static friction.