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A plumb bob of mass 1kg is hung from the ceiling of a train compartment, if the train moves with an acceleration a=g / 2 up the plane. If the angle of incline is
$30 °$ . Find the angle made by the string with the normal to the ceiling. Also, find the tension in the string. (Take, $g = 10 m / s^{2}$ )

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$\sin^{- 1} (\frac{2}{\sqrt{3}}), \frac{\sqrt{7}}{2} m g = 4 \sqrt{7} N$

$\tan^{- 1} (\frac{2}{\sqrt{3}}), \frac{\sqrt{7}}{2} m g = 5 \sqrt{7} N$

$\cot^{- 1} (\frac{2}{\sqrt{3}}), \frac{\sqrt{7}}{2} m g = 4 \sqrt{7} N$

$\cos^{- 1} (\frac{2}{\sqrt{3}}), \frac{\sqrt{7}}{2} m g = 9 \sqrt{7} N$

answer is D.

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Detailed Solution

$T \sin θ - m g \sin 30 ° = m a = \frac{m g}{2}$

$∴ T \sin θ = m g$ …………….(i)

$T \cos θ = m g \cos 30 °$ ………………….(ii)

or $T \cos θ = \frac{\sqrt{3} m g}{2}$

Solving Eqs. (i) and (ii), we get

$θ = \tan^{- 1} (\frac{2}{\sqrt{3}})$

and $T = \frac{\sqrt{7}}{2} m g = 5 \sqrt{7} N$

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