ABSTRACT
At the end of this chapter you should be able to: • understand centripetal force • understand D’Alembert’s principle • understand centrifugal force • solve problems involving locomotives and
cars travelling around bends • solve problems involving a conical
pendulum • solve problems involving the motion in a
vertical circle • understand the centrifugal clutch
16.1 Introduction In this chapter we will restrict ourselves to the uniform circular motion of particles. We will assume that objects such as railway trains and motorcars behave as particles, i.e. rigid body motion is neglected. When a railway train goes round a bend, its wheels will have to produce a centripetal acceleration towards the centre of the turning circle. This in turn will cause the railway tracks to experience a centrifugal thrust, which will tend to cause the track to move outwards. To avoid this unwanted outward thrust on the outer rail, it will be necessary to incline the railway tracks in the manner shown in Figure 16.1. From Section 13.3, it can be seen that when a particle moves in a circular path at a constant speed v, its centripetal acceleration,
a = 2v sin θ 2
× 1 t
When θ is small, θ ≈ sin θ , hence a = 2v θ
2 × 1
t = v θ
t
However, ω = uniform angular velocity = θ t
a
If r = the radius of the turning circle, then v = ωr
and a = ω2r = v 2
r
Now force = mass × acceleration Hence,
centripetal force = mω2r = mv 2
r (16.1)
Although problems involving the motion in a circle are dynamic ones, they can be reduced to static problems through D’Alembert’s principle. In this principle, the centripetal force is replaced by an imaginary centrifugal force which acts equal and opposite to the centripetal force. By using this principle, the dynamic problem is reduced to a static one.
