#### Sample problem: Motion in a horizontal circle

Two particles, A and B, each of mass *m* are connected by a string which passes through a hole in a smooth table. One particle rotates in a horizontal circle on the table, the other remains suspended underneath the table (See Fig. 1). If the suspended particle is to remain at rest, how many revolutions per minute does the particle on the table have to perform in a radius of (a) *r* = 0.1 metres, (b)* r* = 0.5 metres ?

The physical model of the problem is described with the aid of a diagram in Fig. 1:

#### Solution

Fig. 2 show the forces acting in the physical model to help us construct the mathematical equations:

The forces acting in the system are shown in Fig. 2:

*T* = Tension in the string in Newtons

*w* = Angular velocity of rotating particle in radians/second

For particle *A*:* T = mw *^{2}r N

For particle *B*:* T = mg *N

These mathematical equations describe how the physical model will behave.

Therefore: *mg = mw *^{2}r for no movement of particle *B* to occur.

Þ *g = w *^{2}r Þ w ^{2 }= Þ w =

As: 1 revolution = 360^{o} = 2*p *radians

Þ 1 revolution/second = 2*p *radians/second

Þ w * = *2p * f* (where *f* = frequency in revolutions/second)

Þ *f *

(a) *r* = 0.1 m Þ *f* = 1.576 revolutions/second = 94.58 revolutions/minute.

(b) *r* = 0.5 m Þ *f* = 0.705 revolutions/second = 42.298 revolutions/minute.