System of Particles and Rotational Motion - Complete Summary, Formula, Mind Map & Important Questions (Physics)

Physics Smart Booklet




7.System of Particles
and Rotational Motion




Physics Smart Booklet
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System of Particles and
Rotational Dynamics
A group of particles that can be identified and distinguished from other particles or groups is called a system. The
motion of the system as a whole can be analysed by applying the laws of mechanics. This is achieved by using the
concept of ‘centre of mass’. Thus, identifying the centre of mass is quite important in the study of dynamics of a
system.
Rigid body: It is a body whose shape and size do not change during its state of rest or motion.
A classic example of a physical system is that of a rigid body, in which the relative distance between any two particles
remains unaltered during its motion. Study of rigid body motion involves physical parameters like moment of inertia,
torque, angular velocity, angular momentum, translational energy and rotational energy.
Centre of mass of a system of two particles
Centre of mass is the point at which the entire mass of a body is supposed to be concentrated.
If we have discrete system of particles as shown in the figure, then centre of mass is defined as
m R + m 2 R 2 + m3 R 3 + . . .
R cm = 1 1
m1 + m 2 + m3 + . . .
1
R cm =
M
 mi R i
The coordinates of centre of mass
m x + m2 x 2 + . . . 1
Xcm = 1 1
m1 + m2 + m3
=
M
 mi xi
m y + m2 y2 + m3 y3 + . . . 1
Ycm = 1 1 =  mi yi
m1 + m2 + m3 + . . . M
m z + m2 z 2 + . . . 1
Zcm = 1 1 =  mi zi
m1 + m2 + . . . M
For a system having continuous distribution of the mass, the coordinates of cm are
1 1 1
Xcm =  xdm, Ycm =  ydm , Zcm =  zdm
M M M
• Position of centre of mass is independent of coordinate system choosen.
• Centre of mass depends on the shape of the body and distribution of mass.
• Centre of mass coincides with geometric centre bodies where mass is homogeneous.
• Centre of mass remains unchanged in rotatory motion while in translatory motion position changes.
• If small position of mass m2 is removed from a larger position of mass m1. Then centre of mass of the remaining part
m x − m2 x 2
is x cm = 1 1
m1 − m2
Motion of centre of mass
If a system of particles of masses m1, m2, . . move with velocities v1, v2, v3 . . . respectively
• Then velocity of centre of mass is given by
m v + m 2 v 2 + . . . + m n v n  m1vi
VCM = 1 1 =
m1 + m 2 + . . . + m n M
(M → total mass of the body)




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• Momentum of the centre of mass is
Mvcm = m1v1 + m 2 + v 2 + . . . + m n v n
PCM = P1 + P2 + . . . + Pn
If VCM = 0, PCM = 0 i.e., in the frame of reference of CM, the momentum of a system is zero.
• Acceleration of CM is given by
m1a1 + m 2 a 2 + . . . m n a n  mi a i
a CM = =
m1 + m 2 + . . . m n M
Or Ma CM = m1a1 + m2 a 2 + . . . m n a n
Fext = F1 + F2 + . . . + Fn
This is equation of motion of centre of mass.
If Fext = 0  a CM = 0  vCM = constant.
If Fext = 0, no external force acts on a system, then the velocity of its CM remains constant.
 velocity of CM is not affected by internal forces.
• If Fext = 0  a CM = 0  VCM = constant, then p = constant.
This leads to conservation of linear momentum.
• If a system of 2 particles of mass m1 and m2 separated by a distance x initially at rest, moving towards each other
under the action of attractive force then the 2 particles collide at their centre of mass.
a m
Here F12 = − F21 or m1a1 = m2a2  1 = 2
a 2 m1
Since initial momentum = 0, centre of mass is at rest.
v m
VCM = 0, m1v1 = m2 v2 or 1 = 2
v2 m1
x1 m2
 Ratio of distances covered by particles before collision is =
x 2 m1
Rotational motion: A rigid body undergoes rotational motion when each of its particles travel in a circle centered
on a straight line, called the axis of rotation
Rotational variables: The rotational variables are the angular equivalents of the
y
linear quantities position, displacement, velocity and acceleration
Lightly
Rotation axis




Angular position (): It is the position of a fixed line perpendicular to the axis of shaded
Reference
rotation, fixed in the body, relative to a fixed axis. It is also called the angular line

s 
coordinate. = where Z
x
r A rigid body rotating
anticlockwise in the
s → arc length described by a point on the reference line relative to the fixed axis. xy- phase about z axis
the rotation axis
r → radius of the arc. Its SI unit is the radian (rad) without any dimensions.
Angular displacement: It is the difference in the angular coordinates of rotating body at times t1 and
t2 = t1 + t
 = 2 − 1




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