Question 42: Laws of Motion

Question

A flexible chain of mass $m$ hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is $30^{\circ}$. Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is $\_\_\_\_$ .

Accepted Answer

Considering the left half of the chain, the mass of this segment is $\frac{m}{2}$.
Weight ( $\mathrm{W}_1$ ) acts vertically downward through the centre of gravity of the half-segment.
$$ \mathrm{W}_1=\frac{\mathrm{mg}}{2} $$
Tension at support ( $\mathrm{T}_2$ ) acts at an angle of $30^{\circ}$ to the horizontal.
Tension at lowest point $\left(\mathrm{T}_1\right)$ at the lowest point, the chain is perfectly horizontal, so the tension $\mathrm{T}_1$ acts purely in the horizontal direction.
For the half-chain to be in equilibrium, the sum of vertical and horizontal forces must be zero.
The vertical component of the support tension must balance the weight of the half-chain.
$\mathrm{T}_2 \sin \left(30^{\circ}\right)=\frac{\mathrm{mg}}{2}$
$$\Rightarrow $$ $$ \mathrm{T}_2\left(\frac{1}{2}\right)=\frac{\mathrm{mg}}{2} \Rightarrow \mathrm{~T}_2=\mathrm{mg} \ldots (i)$$
The horizontal component of the support tension must balance the tension at the lowest point.
$$ \mathrm{T}_1=\mathrm{T}_2 \cos \left(30^{\circ}\right) $$
From equation (i)
$$ \mathrm{T}_1=(\mathrm{mg}) \cos \left(30^{\circ}\right)=\frac{\sqrt{3}}{2} \mathrm{mg} $$
Therefore, the tension at the lowest point of the chain is $\frac{\sqrt{3}}{2} \mathrm{mg}$.
Hence, the correct option is (B).

Answer

$m \mathrm{~g}$

Answer

$\frac{\sqrt{3}}{2} m g$

Answer

$\frac{1}{2} m g$

Answer

$\sqrt{3} m \mathrm{~g}$

Laws of Motion

A flexible chain of mass m hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is 30^{\circ}. Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is \_\_\_\_ .

Choose the correct answer: