Pearson Edexcel IAL Mathematics M3.4.4 Motion of a particle in a vertical circle

Practise vertical circle problems using energy, radial equations, tension and contact conditions at key positions.

Exam points

use energy to link speed changes with height around a vertical circle
form radial equations for tension, reaction or weight at named positions
test complete-circle or slack-string conditions using v^2 and force limits

[Maximum number: 4]

Figure 5

A thin smooth hollow spherical shell has centre O and radius r. Part of the shell is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point A is the lowest point of the bowl, as shown in Figure 5.

The point B is on the rim of the bowl, with OB at an angle $\theta$ to the upward vertical, where $\tan \theta=\frac{12}{5}$
A small ball is placed in the bowl at A. The ball is projected from A with horizontal speed u and moves in the vertical plane AOB. The ball stays in contact with the bowl until it reaches B.

At the instant when the ball reaches B, the speed of the ball is v.
By modelling the ball as a particle and ignoring air resistance,

show that $u^{2} \geqslant \frac{41}{13} g r$

\frac{mv^2}{r}\leq mg\cos\theta

v^2\leq gr\cos\theta=\frac{5gr}{13}

u^2-\frac{36gr}{13}\leq \frac{5gr}{13}

A1*

u^2\geq \frac{41gr}{13}

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M3.4.4 - Motion of a particle in a vertical circle question 1