First slide

Vertical circular motion

Question

A mass m is revolving in a vertical circle at the end of a string of length 20 cm. By how much does the tension of the string at the lowest point exceed the tension at the top most point?

Moderate

A

2 mg
B

4 mg
C

6 mg
D

8 mg

Solution

The tension T₁ at the topmost point is given by
$T_{1} = \frac{{mv}_{1}^{2}}{20} - mg$
Centrifugal force acting outward while weight acting downward.
The tension T₂ at the lowest point
$T_{2} = \frac{{mv}_{2}^{2}}{20} + mg$
centrifugal force and weight (both) acting downward
$\begin{array}{l} T_{2} - T_{1} = \frac{{mv}_{2}^{2} - {mv}_{1}^{2}}{20} + 2 mg \\ v_{1}^{2} = v_{2}^{2} - 2 gh \\ or v_{2}^{2} - v_{1}^{2} = 2 g (40) = 80 g \\ ∴ T_{2} - T_{1} = \frac{80 mg}{20} + 2 mg = 6 mg \end{array}$

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