Important Formulas: System of Particles and Rotational Motion | Physics Class 11 - NEET PDF Download

 Page 2


 
 
? A triangular plate (By qualitative argument)
at the centroid : y
c
 = 
3
h
? A semi-circular ring   y
c
 = 
?
R 2
    x
c
 = O
? A semi-circular disc y
c
 = 
? 3
R 4
   x
c
 = O
? A hemispherical shell    y
c
 = 
2
R
   x
c
 = O
? A solid hemisphere   y
c
 = 
8
R 3
  x
c
 = O
? A circular cone (solid) y
c
 = 
4
h
? A circular cone (hollow) y
c
 = 
3
h
Page 3


 
 
? A triangular plate (By qualitative argument)
at the centroid : y
c
 = 
3
h
? A semi-circular ring   y
c
 = 
?
R 2
    x
c
 = O
? A semi-circular disc y
c
 = 
? 3
R 4
   x
c
 = O
? A hemispherical shell    y
c
 = 
2
R
   x
c
 = O
? A solid hemisphere   y
c
 = 
8
R 3
  x
c
 = O
? A circular cone (solid) y
c
 = 
4
h
? A circular cone (hollow) y
c
 = 
3
h
MOTION OF CENTRE OF MASS AND CONSERVA TION OF MOMENTUM:
Velocity of centre of mass of system
cm
v
?
= 
M
dt
dr
m .... ..........
dt
dr
m
dt
dr
m
dt
dr
m
n
n
3
3
2
2
1
1
? ? ?
= 
M
v m .......... v m v m v m
n n 3 3 2 2 1 1
? ? ? ?
? ? ?
System P
 = M cm v
Acceleration of centre of mass of system
cm
a
?
 = 
M
dt
dv
m .... ..........
dt
dv
m
dt
dv
m
dt
dv
m
n
n
3
3
2
2
1
1
? ? ?
       = 
M
a m .......... a m a m a m
n n 3 3 2 2 1 1
? ? ? ?
? ? ?
       = 
M
system on force Net
 = 
M
Force ernal int Net Force External Net ?
       = 
M
Force External Net
ext
F
?
 = M
cm
a
?
IMPULSE
Impulse of a force F action on a body is defined as :-
 
J
?
 = 
?
f
i
t
t
Fdt
P ? J
? ?
?
(impulse - momentum theorem)
Important points :
1. Gravitational force and spring force are always non-impulsive.
2. An impulsive force can only be balanced by another impulsive force.
COEFFICIENT OF RESTITUTION (e)
e = 
n deformatio of pulse Im
n reformatio of pulse Im
 = 
?
?
dt F
dt F
d
r
= s
impact of line along approach of Velocity
impact of line along separation of Velocity
Page 4


 
 
? A triangular plate (By qualitative argument)
at the centroid : y
c
 = 
3
h
? A semi-circular ring   y
c
 = 
?
R 2
    x
c
 = O
? A semi-circular disc y
c
 = 
? 3
R 4
   x
c
 = O
? A hemispherical shell    y
c
 = 
2
R
   x
c
 = O
? A solid hemisphere   y
c
 = 
8
R 3
  x
c
 = O
? A circular cone (solid) y
c
 = 
4
h
? A circular cone (hollow) y
c
 = 
3
h
MOTION OF CENTRE OF MASS AND CONSERVA TION OF MOMENTUM:
Velocity of centre of mass of system
cm
v
?
= 
M
dt
dr
m .... ..........
dt
dr
m
dt
dr
m
dt
dr
m
n
n
3
3
2
2
1
1
? ? ?
= 
M
v m .......... v m v m v m
n n 3 3 2 2 1 1
? ? ? ?
? ? ?
System P
 = M cm v
Acceleration of centre of mass of system
cm
a
?
 = 
M
dt
dv
m .... ..........
dt
dv
m
dt
dv
m
dt
dv
m
n
n
3
3
2
2
1
1
? ? ?
       = 
M
a m .......... a m a m a m
n n 3 3 2 2 1 1
? ? ? ?
? ? ?
       = 
M
system on force Net
 = 
M
Force ernal int Net Force External Net ?
       = 
M
Force External Net
ext
F
?
 = M
cm
a
?
IMPULSE
Impulse of a force F action on a body is defined as :-
 
J
?
 = 
?
f
i
t
t
Fdt
P ? J
? ?
?
(impulse - momentum theorem)
Important points :
1. Gravitational force and spring force are always non-impulsive.
2. An impulsive force can only be balanced by another impulsive force.
COEFFICIENT OF RESTITUTION (e)
e = 
n deformatio of pulse Im
n reformatio of pulse Im
 = 
?
?
dt F
dt F
d
r
= s
impact of line along approach of Velocity
impact of line along separation of Velocity
(a) e = 1 ? Impulse of Reformation  = Impulse of Deformation
? V elocity of separation = V elocity of approach
? Kinetic Energy may be conserved
? Elastic collision.
(b) e = 0 ? Impulse of Reformation = 0
? V elocity of separation = 0
? Kinetic Energy is not conserved
? Perfectly Inelastic collision.
(c) 0 < e < 1 ? Impulse of Reformation  < Impulse of Deformation
? V elocity of separation < V elocity of approach
? Kinetic Energy is not conserved
? Inelastic collision.
VARIABLE MASS SYSTEM :
If a mass is added or ejected from a system, at rate ? kg/s and relative
velocity 
rel
v
?
 (w.r.t. the system), then the force exerted by this mass
on the system has magnitude 
rel
v
?
? .
Thrust Force (
t
F
?
)
?
?
?
?
?
?
?
dt
dm
v F
rel t
?
?
Rocket propulsion :
If gravity is ignored and initial velocity of the rocket u = 0;
v = v
r
 ln ?
?
?
?
?
?
m
m
0
.
RIGID BODY DYNAMICS
1. RIGID BODY :
A
V
A
V
B
?
2
?
1
B
V
B
sin ?
2
V
A
cos ?
1
V
A
sin ?
1
V
B
cos ?
2
A
B
Page 5


 
 
? A triangular plate (By qualitative argument)
at the centroid : y
c
 = 
3
h
? A semi-circular ring   y
c
 = 
?
R 2
    x
c
 = O
? A semi-circular disc y
c
 = 
? 3
R 4
   x
c
 = O
? A hemispherical shell    y
c
 = 
2
R
   x
c
 = O
? A solid hemisphere   y
c
 = 
8
R 3
  x
c
 = O
? A circular cone (solid) y
c
 = 
4
h
? A circular cone (hollow) y
c
 = 
3
h
MOTION OF CENTRE OF MASS AND CONSERVA TION OF MOMENTUM:
Velocity of centre of mass of system
cm
v
?
= 
M
dt
dr
m .... ..........
dt
dr
m
dt
dr
m
dt
dr
m
n
n
3
3
2
2
1
1
? ? ?
= 
M
v m .......... v m v m v m
n n 3 3 2 2 1 1
? ? ? ?
? ? ?
System P
 = M cm v
Acceleration of centre of mass of system
cm
a
?
 = 
M
dt
dv
m .... ..........
dt
dv
m
dt
dv
m
dt
dv
m
n
n
3
3
2
2
1
1
? ? ?
       = 
M
a m .......... a m a m a m
n n 3 3 2 2 1 1
? ? ? ?
? ? ?
       = 
M
system on force Net
 = 
M
Force ernal int Net Force External Net ?
       = 
M
Force External Net
ext
F
?
 = M
cm
a
?
IMPULSE
Impulse of a force F action on a body is defined as :-
 
J
?
 = 
?
f
i
t
t
Fdt
P ? J
? ?
?
(impulse - momentum theorem)
Important points :
1. Gravitational force and spring force are always non-impulsive.
2. An impulsive force can only be balanced by another impulsive force.
COEFFICIENT OF RESTITUTION (e)
e = 
n deformatio of pulse Im
n reformatio of pulse Im
 = 
?
?
dt F
dt F
d
r
= s
impact of line along approach of Velocity
impact of line along separation of Velocity
(a) e = 1 ? Impulse of Reformation  = Impulse of Deformation
? V elocity of separation = V elocity of approach
? Kinetic Energy may be conserved
? Elastic collision.
(b) e = 0 ? Impulse of Reformation = 0
? V elocity of separation = 0
? Kinetic Energy is not conserved
? Perfectly Inelastic collision.
(c) 0 < e < 1 ? Impulse of Reformation  < Impulse of Deformation
? V elocity of separation < V elocity of approach
? Kinetic Energy is not conserved
? Inelastic collision.
VARIABLE MASS SYSTEM :
If a mass is added or ejected from a system, at rate ? kg/s and relative
velocity 
rel
v
?
 (w.r.t. the system), then the force exerted by this mass
on the system has magnitude 
rel
v
?
? .
Thrust Force (
t
F
?
)
?
?
?
?
?
?
?
dt
dm
v F
rel t
?
?
Rocket propulsion :
If gravity is ignored and initial velocity of the rocket u = 0;
v = v
r
 ln ?
?
?
?
?
?
m
m
0
.
RIGID BODY DYNAMICS
1. RIGID BODY :
A
V
A
V
B
?
2
?
1
B
V
B
sin ?
2
V
A
cos ?
1
V
A
sin ?
1
V
B
cos ?
2
A
B
If the above body is rigid
V
A
 cos
 
?
1
   =  V
B
 cos
 
?
2
V
BA
 = relative velocity of point B with respect to point A.
V
BA
B
A
    
Pure Translational
         Motion
Pure Rotational 
         Motion
Types of Motion of rigid body
Combined Translational and 
Rotational Motion
2. MOMENT OF INERTIA (I) :
Definition : Moment of Inertia is defined as the capability of system
to oppose the change produced  in the rotational motion of a body.
Moment of Inertia is a scalar positive quantity.
? = mr
1
2
 + m
2 
r
2
2 
+.........................
  = ?
?
 + ?
?
 + ?
?
  
+.........................
S ? units of Moment of Inertia is Kgm
2
.
Moment of Inertia of  :
2.1 A single particle  : ? = mr
2
where  m = mass of the particle
r = perpendicular distance of the particle from the axis about
which moment of Inertia is to be calculated
2.2 For many particles (system of particles)  :
? = ?
?
n
1 i
2
i i
r m
2.3 For a continuous object :
? =  
?
2
dmr
where  dm = mass of a small element
r = perpendicular distance of the particle from the axis