Question

A body of mass $m$ is suspended by two strings making angles $\theta_{1}$ and $\theta_{2}$ with the horizontal ceiling with tensions $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$ simultaneously. $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$ are related by $\mathrm{T}_{1}=\sqrt{3} \mathrm{~T}_{2}$. the angles $\theta_{1}$ and $\theta_{2}$ are

• A. $\theta_{1}=30^{\circ} \theta_{2}=60^{\circ}$ with $\mathrm{T}_{2}=\frac{3 \mathrm{mg}}{4}$
• B. $\theta_{1}=60^{\circ} \theta_{2}=30^{\circ}$ with $\mathrm{T}_{2}=\frac{\mathrm{mg}}{2}$
• C. $\theta_{1}=45^{\circ} \theta_{2}=45^{\circ}$ with $\mathrm{T}_{2}=\frac{3 \mathrm{mg}}{4}$
• D. $\theta_{1}=30^{\circ} \theta_{2}=60^{\circ}$ with $\mathrm{T}_{2}=\frac{4 \mathrm{mg}}{5}$

Hint:

Use the equilibrium condition to find the tensions and angles.

Answer 1

The Correct Option is B

1. Given the equations: \[ \mathrm{T}_{1} \sin \theta_{1} + \mathrm{T}_{2} \sin \theta_{2} = \mathrm{mg} \] \[ \mathrm{T}_{1} = \sqrt{3} \mathrm{~T}_{2} \]
2. Substitute $\mathrm{T}_{1}$: \[ \sqrt{3} \mathrm{~T}_{2} \sin \theta_{1} + \mathrm{T}_{2} \sin \theta_{2} = \mathrm{mg} \] \[ \mathrm{T}_{2} (\sqrt{3} \sin \theta_{1} + \sin \theta_{2}) = \mathrm{mg} \]
3. For $\theta_{1} = 60^{\circ}$ and $\theta_{2} = 30^{\circ}$: \[ \mathrm{T}_{2} (\sqrt{3} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2}) = \mathrm{mg} \] \[ \mathrm{T}_{2} = \frac{\mathrm{mg}}{2} \] Therefore, the solution with $\theta_{1}=60^{\circ}$ and $\theta_{2}=30^{\circ}$ yields $\mathrm{T}_{2}=\frac{\mathrm{mg}}{2}$, corresponding to option (2).