(^) Ndt mv mv ' (i)
Angular impulse by friction in the vertical direction
(^22)
5
fR R Ndt mR v
R
(ii)From eqs. (i) and (ii), we get(^) Ndt mVand v v^2 '
4.Sol: Let v be the velocity of the centre of mass of
the sphere and be the angular velocity of the
body about an axis passing through the centre of
mass.
The linear impulse is
J = Mv
The angular impulse is
^2
2
5
J h R MR
From the above two equations, ^2
2
5
v h R R
From the condition of pure rolling, v R
2 7
5 5
h R h R R
5.Sol: Given system of two particles will rotate about
its centre of mass.
Initial angular momentum
2
Mv L
Final angular momentum
2
2
2
L
M
From conservation of angular momentum
2
2
2 2
L L v
Mv M
L
6.Sol: From linear impluse
J = mv (v-velocity of CM)
From angular impluse
2
12
mL
Jx
7.Sol: (^) i
As the point P velocity is zero.
0
P 2
l
v v
(^2)
l
v
From the above three eq’s 6
xL
L I mvR
L I mRf ^2 '
(^) f i ' 2
L L I mvR
I mR
8.Sol: The two discs exert equal and opposite forces
on each other when in contact. The torque due to
these forces changes the angular momentum of
each disc. Let 1 and 2 are the angular velocities
of the two discs.
4.Sol:
5.Sol:
6.Sol:
7.Sol:
8.Sol:
The angular impulse on the two discs are
fa t I 1 0 1 (i)
and fb t I 2 2 (ii)
From eqns. (i) and (ii), we get
1 0 1
2 2
a I
b I
(iii)
When slipping ceases between the discs, the
contact points of the two discs have the same linear
velocity, i.e.,
a b 1 2 (iv)
On substituting 2 in eq’s (iii), we get
1 0
(^122)
1 2 /
I
I a I b