94
2 Particle Dynamics
If n particles fall per second, the force exerted on the pan is
F = +2mnv = (2)(0.1)
441
60
(5.6) = 8.232 N
=
8.232
9.8
kg wt = 0.84 kg wt
2.57 In this case, the particles will stick to the pan. Therefore the scale reading will
increase due to the weight of the particles that get accumulated in the pan.
For complete inelastic collision p = mv as the final momentum is zero. Net
force on the scale = weight of the particle + force of impact. At time t, scale
reading (in newtons)
= mngt + mn
2gh
= mng
t +
2h
g
Scale reading in kg wt = mn
t +
2h
g
2.58 Let a sphere of mass m 1 travelling with velocity u 1 collide with the second
sphere of mass m 2 at rest, with their centres in straight line. After the collision
let the final velocities be v 1 and v 2 , respectively, for m 1 and m 2 . By definition
the coefficient of restitution e is given by the ratio
e =
Relative velocity of separation
Relative velocity of approach
=
v 2 − v 1
v 1
(1)
Momentum conservation requires that total momentum before collision =
total momentum after collision:
m 1 u 1 = m 1 v 1 + m 2 v 2
(2)
Eliminating v 2 between (1) and (2),
v 1 =
(m 1 − em 2 )u 1
m 1 + m 2
(3)
v 2 =
m 1 (1 + e)u 1
m 1 + m 2
(4)
(i) Putting u 1 = u, m 1 = m and m 2 =
m
2
v 1 =
u
3
(2 − e)
(5)
v 2 =
2u
3
(1 + e)
(6)