Figure below shows a small mass connected to a string, which is attached to a vertical post. If the ball is released when the string is horizontal as shown, the magnitude of the total acceleration (including radial and tangential) of the mass as a function of the angle $\theta$ is- Position

• A

  $g \sin \theta$
• B

  $g \sqrt{3 \cos ^{2} \theta+1}$
• C

  $g \cos \theta$
• Correct answer
  D

  $g \sqrt{3 \sin ^{2} \theta+1}$

Solution

Radical acceleration $=\frac{\mathrm{V}^{2}}{\mathrm{R}}=\frac{2 \mathrm{~g} \ell \sin \theta}{\ell}=2 \mathrm{~g} \sin \theta$ tangential acceleration $=g \cos \theta$ total acceleration $=\sqrt{4 \mathrm{~g}^{2} \sin ^{2} \theta+\mathrm{g}^{2} \cos ^{2} \theta} \Rightarrow \mathrm{g} \sqrt{1+3 \sin ^{2} \theta}$

Hence, the correct answer is option: 'D'.