Environmental Engineering Reference
In-Depth Information
angular velocity. The torque is a result of the friction acting at the point of contact
between the ball and the horizontal surface and has magnitude FR
=
µ
k
MgR,
which causes an increase in ω. Hence
I
C
d
ω
=
=
τ
C
µ
k
MgR
d
t
.
Integrating and using the condition ω
=
0
at t
=
0
we obtain
µ
k
MgRt
I
C
ω
=
.
(4.42)
When does this skidding phase end? At some time t
r
the ball will start to roll. At
this time the friction drops to almost zero since the surfaces of the ball and the floor
no longer slide past each other. For rolling to occur the instantaneous velocity of
the point of contact with the floor must be zero, i.e. the sum of the centre-of-mass
velocity and the velocity due to rotation about the centre of mass must be zero:
0
=
V
C
−
Rω,
which gives
V
C
=
Rω.
(4.43)
Eq. (4.43) is the rolling condition. We may use it to express ω in terms of V
C
in
Eq. (4.42) when t
=
t
r
. We obtain
µ
k
MgR
2
t
r
I
C
=
V
0
−
µ
k
gt
r
,
which when rearranged gives
V
0
µ
k
g
1
t
r
=
.
MR
2
I
C
+
We can also use Eq. (4.41) to deduce V
r
, the speed at which rolling begins:
V
0
=
V
r
.
I
C
MR
2
1
+
For t>t
r
there is much less friction and as a result a much reduced force acting
on the ball. The ball's velocity will therefore be approximately constant and so V
r
is approximately the final speed of the ball.
4.5 ANGULAR IMPULSE AND THE CENTRE OF PERCUSSION
Have you ever wondered why a cricket bat, a baseball bat or a tennis racquet has
a sweet spot? This is the point on the bat where the ball seems to be hit most cleanly,
