First slide

Mechanical equilibrium force cases

Question

A ball of mass 1 kg hangs in equilibrium from two strings OA and OB as shown in figure. The ratio of tensions in strings OA and OB is : $(Take g = 10 {ms}^{- 2})$

Moderate

A

$\sqrt{3}$
B

$\frac{\sqrt{3}}{2}$
C

$\frac{1}{\sqrt{3}}$
D

$\frac{1}{2 \sqrt{3}}$

Solution

Forces acting on the ball are as shown in figure.
The three concurrent forces are in equilibrium
Using Lami's theorem.
$\frac{T_{1}}{\sin 150^{\circ}} = \frac{T_{2}}{\sin 120^{\circ}} = \frac{10}{\sin 90^{\circ}}$
$\Rightarrow \frac{T_{1}}{\sin 30^{\circ}} = \frac{T_{2}}{\sin 60^{\circ}} = \frac{10}{1} \Rightarrow \frac{T_{1}}{T_{2}}$
$∴ T_{1} = 10 \sin 30^{\circ} = 10 \times 0 .5 = 5 N$ and T₂ = 10
$\sin 60^{\circ} = 10 \times \frac{\sqrt{3}}{2} = 5 \sqrt{3} N$

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