122
3 Rotational Kinematics
Fig. 3.14
centre of gravity produces a clockwise torque. The condition for the maximum
speed v max is given by equating these two torques:
mv 2
max
r
h = mga
or v max =
gra
h
=
9.8 × 100 × 0.75
1.0
= 27.11 m/s
3.23 (a) When a vehicle takes a turn on a level road, the necessary centripetal force
is provided by the friction between the tyres and the road. However, this
results in a lot of wear and tear of tyres. Further, the frictional force may
not be large enough to cause a sharp turn on a smooth road.
If the road is constructed so that it is tilted from the horizontal, the road
is said to be banked. Figure 3.15 shows the profile of a banked road at an
angle θ with the horizontal. The necessary centripetal force is provided
by the horizontal component of the normal reaction N and the horizontal
component of frictional force.
Three external forces act on the vehicle, and they are not balanced, the
weight W , the normal reaction N , and the frictional force. Balancing the
horizontal components
mv 2
r
= μ mg cos
2
θ + N sin θ
or N sin θ =
mv 2
r
− μ mg cos
2
θ
(1)
Balancing the vertical components
mg = N cos θ − μmg cos θ sin θ
or N cos θ = mg + μmg cos θ sin θ
(2)