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Question
A simple pendulum consists of a 50 cm long string connected to a 100 g ball. The ball is pulled aside so that the string makes an angle of 37° with the vertical and is then released. Find the tension in the string when the bob is at its lowest position.
Solution
From the figure,
\[\cos \theta = \frac{\text{OC}}{\text{OB}}\]
\[ \Rightarrow OC = OB \cos \theta\]
\[ = \left( 0 . 5 \right) \times \left( 0 . 8 \right) = 0 . 4\]
\[\text{ So }, CA = \left( 0 . 5 \right) - \left( 0 . 4 \right)\]
\[ = 0 . 1 \text{ m}\]
Total energy at A = Total energy at B
\[\frac{1}{2}\text{m} \nu^2 = \text{mg} \left( AC \right)\]
\[ \nu^2 = 2 \times 10 \times \left( 0 . 1 \right) = 2\]
\[ \Rightarrow \text{v} = \sqrt{2}\]
So, the tension is given by
\[\text{T} = \frac{\text{m} \nu^2}{r} + \text{mg}\]
\[ = \left( 0 . 1 \right) \left( \frac{2}{0 . 5} + 10 \right) = 1 . 4 \text{ N }\]
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