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Q.

The given figure shows a small mass connected to a string, which is attached to a vertical post. If the mass is released from rest when the string is horizontal as shown, the magnitude of the total acceleration of the mass as a function of the angle $θ$ is

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a

$2 g \sin θ$

b

$2 g \cos θ$

c

$g \sqrt{3 \cos^{2} θ + 1}$

d

$g \sqrt{3 \sin^{2} θ + 1}$

answer is D.

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Detailed Solution

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From conservation of energy $Δ K + Δ U = 0$

$\begin{matrix} mgh = \frac{1}{2} m v^{} \Rightarrow^{} mglsinθ = \frac{1}{2} {mv}^{2} \\ \Rightarrow a_{C} = 2 gsin θ = \frac{v^{2}}{l} = radial acceleration \end{matrix}$

$gcos θ = a_{t} = tangential acceleration$

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Total acceleration

$\begin{aligned} a = \sqrt{a_{c}^{2} + a_{t}^{2}} = g \sqrt{\cos^{2} θ + (2 \sin θ)^{2}} \\ = g \sqrt{(1 - \sin^{2} θ) + 4 \sin^{2} θ} = g \sqrt{1 + 3 \sin^{2} θ} \end{aligned}$

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