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Question
A simple pendulum is suspended from the ceiling of a car taking a turn of radius 10 m at a speed of 36 km/h. Find the angle made by he string of the pendulum with the vertical if this angle does not change during the turn. Take g = 10 m/s2.
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Solution
Speed of the car = v = 36 km/hr = 10 m/s
Acceleration due to gravity = g = 10 m/s2
Let T be the tension in the string when the pendulum makes an angle θ with the vertical.
From the figure, we get :
\[\text{T}\sin\theta = \frac{\text{mv}^2}{r} . . . (\text{i})\]
\[T\cos\theta = \text{mg . . . (ii)}\]
\[ \Rightarrow \frac{\sin\theta}{\cos\theta} = \frac{\text{mv}^2}{\text{rmg}}\]
\[ \Rightarrow \tan\theta = \frac{\text{v}^2}{\text{rg}}\]
\[ \Rightarrow \theta = \tan^{- 1} \left( \frac{\text{v}^2}{\text{rg}} \right)\]
\[ = \tan^{- 1} \left[ \frac{100}{(10 \times 10)} \right]\]
\[ = \tan^{- 1} (1)\]
\[ \Rightarrow \theta= 45^\circ\]
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