First slide

Conservation of mechanical energy

Question

For a particle rotating in a vertical circle with uniform speed, the maximum and minimum magnitudes of external force (other than gravity) acting on the particle are in the ratio 5:3. If the radius of vertical circle is 2m, the speed of revolving body (in m/s) is_____ (take $g = 10 m / s^{2}$ )

Moderate

Solution

Maximum tension at bottom most point is given by
$T_{1} = m g + \frac{m v^{2}}{2} .... (1) ∵ R = 2 m$
Minimum tension at top most point is given by
$T_{2} = \frac{m v^{2}}{2} - m g ..... (2)$
$\begin{array}{l} \frac{T_{1}}{T_{2}} = \frac{5}{3} \\ \Rightarrow \frac{2 m g + m v^{2}}{m v^{2} - 2 m g} = \frac{5}{3} \\ \Rightarrow 6 g + 3 v^{2} = 5 v^{2} - 10 g \\ \Rightarrow 16 g = 2 v^{2} \\ ∴ v^{2} = 80 \\ ∴ v = \sqrt{80} = 8.94 \end{array}$

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