A Classical Approach of Newtonian Mechanics

7 CIRCULAR MOTION 7.7 Motion on curved surfaces

v’ (^) R
B
mg cos 
mg v
 r cos (^) 
r
A
mg
Figure 65: Motion on the inside of a vertical hoop.

which reduces to

vJ^2 = v^2 − 2 r g (1 − cos θ). (7.55)

Here, v is the velocity at point A (θ = 0 ◦), and vJ is the velocity at point B

(θ = θ◦).

Let us now examine the radial acceleration of the object at point B. The radial

forces acting on the object are the reaction R of the vertical hoop, which acts

towards the centre of the hoop, and the component m g cos θ of the object’s
weight, which acts away from the centre of the hoop. Since the object is executing

circular motion with instantaneous tangential velocity vJ, it must experience an

instantaneous acceleration vJ^2 /r towards the centre of the hoop. Hence, Newton’s
second law of motion yields

m vJ^2

r

= R − m g cos θ. (7.56)

Note, however, that there is a constraint on the reaction R that the hoop can

exert on the object. This reaction must always be positive. In other words, the

hoop can push the object away from itself, but it can never pull it towards itself.

Another way of putting this is that if the reaction ever becomes negative then