Subjective Type

A car is speeding up on a horizontal road with an acceleration $$\alpha$$. Consider the following situations in the car. (i) A ball is suspended from the ceiling through a string and is maintaining a constant angle with the vertical. Find this angle. (ii) A block is kept on a smooth incline and does not slip on the incline. Find the angle of the incline with the horizontal.

Solution

Suppose pendulum makes $$\theta$$ angle with the vertical.
Let $$m$$ be the mass of the pendulum

From the free body diagram
$$T\cos\theta-mg=0$$
$$T\cos\theta=mg$$
$$T=\dfrac{mg}{\cos\theta}..........(i)$$

$$T\sin\theta-ma=0$$
$$T\sin\theta=ma$$
$$T=\dfrac{ma}{\sin\theta}..........(ii)$$

From (i) and (ii)
$$\dfrac{mg}{\cos\theta}=\dfrac{ma}{\sin\theta}$$
$$\tan\theta=\dfrac{a}{g}$$
$$\theta=\tan^{-1}\dfrac{a}{g}$$

Thus angle is $$\tan^{-1}\dfrac{a}{g}$$ with vertical

(ii) Suppose the angle of incline is $$\theta$$
From the diagram
$$ma\cos\theta-mg\sin\theta=0$$
$$ma\cos\theta=mg\sin\theta$$
$$\tan\theta=\dfrac{a}{g}$$

$$\theta=\tan^{-1}\dfrac{a}{g}$$

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