Equilibrium of a Particle questions | Equilibrium of a Particle jee questions

Q 1 :

A 1 kg mass is suspended from the ceiling by a rope of length 4 m. A horizontal force 'F' is applied at the midpoint of the rope so that the rope makes an angle of 45° with respect to the vertical axis as shown in the figure. The magnitude of F is: [2024]

$\frac{10}{\sqrt{2}} N$
1 N
$\frac{1}{10 \times \sqrt{2}} N$
10 N

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(4)

As the system is in equilibrium, therefore

$T_{1} \cos 45 ° = m g$

$T_{1} \sin 45 ° = F$

$∴$ $\tan 45 ° = \frac{F}{m g} \Rightarrow F = m g = 10 N$

Q 2 :

A body of mass 1 kg is suspended with the help of two strings making angles as shown in figure. Magnitude of tensions $T_{1}$ and $T_{2}$ , respectively, are (in N): [2025]

$5, 5 \sqrt{3}$
$5 \sqrt{3}, 5$
$5 \sqrt{3}, 5 \sqrt{3}$
5, 5

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(2)

$T_{1} \sin 60 ° + T_{2} \sin 30 ° = m g$ ... (i)

$T_{1} \cos 60 ° = T_{2} \cos 30 °$ ... (ii)

$T_{1} = \sqrt{3} T_{2}$ ... (iii)

From (i) and (iii)

$\sqrt{3} T_{2} \times \frac{\sqrt{3}}{2} + \frac{T_{2}}{2} = m g$

$T_{2} = \frac{m g}{2} = 5 N$

and $T_{1} = \frac{\sqrt{3} m g}{2} = 5 \sqrt{3} N$

Q 3 :

A body of mass m is suspended by two strings making angles $θ_{1}$ and $θ_{2}$ with the horizontal ceiling with tensions $T_{1}$ and $T_{2}$ simultaneously. $T_{1}$ and $T_{2}$ are related by $T_{1} = \sqrt{3} T_{2}$ . The angle $θ_{1}$ and $θ_{2}$ are [2025]

$θ_{1} = 30 °, θ_{2} = 60 ° with T_{2} = \frac{3 m g}{4}$
$θ_{1} = 60 °, θ_{2} = 30 ° with T_{2} = \frac{m g}{2}$
$θ_{1} = 45 °, θ_{2} = 45 ° with T_{2} = \frac{3 m g}{4}$
$θ_{1} = 30 °, θ_{2} = 60 ° with T_{2} = \frac{4 m g}{5}$

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(2)

$T_{1} \sin θ_{1} + T_{2} \sin θ_{2} = m g$

$T_{1} \cos θ_{1} = T_{2} \cos θ_{2}$

$\sqrt{3} T_{2} \cos θ_{1} = T_{2} \cos θ_{2}$

$\sqrt{3} \cos θ_{1} = \cos θ_{2}$

For $θ_{1} = 60 °, θ_{2} = 30 °$

$\sqrt{3} T_{2} \cdot \frac{\sqrt{3}}{2} + T_{2} \cdot \frac{1}{2} = m g \Rightarrow T_{2} = \frac{m g}{2}$

Q 4 :

A block of $\sqrt{3}$ kg is attached to a string whose other end is attached to the wall. An unknown force F is applied so that the string makes an angle of 30° with the wall. The tension T is (Given g = 10 ${ms}^{- 2}$ ) [2023]

20 N
25 N
10 N
15 N

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(1)

$\cos θ = \frac{\sqrt{3} g}{T}$

$\Rightarrow \frac{\sqrt{3}}{2} = \frac{\sqrt{3} g}{T}$

$\Rightarrow T = 20 N$

Q 5 :

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 $30 °$ . Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is ________. [2026]

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

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(3)

[IMAGE 253]

$T \sin 30 ° = \frac{m}{2} g$

$T \cos 30 ° = T_{0}$

$\tan 30 ° = \frac{m g}{2 T_{0}}$

$T_{0} = \frac{\sqrt{3}}{2} m g$

Topic Question Set

Q 1 :

A 1 kg mass is suspended from the ceiling by a rope of length 4 m. A horizontal force 'F' is applied at the midpoint of the rope so that the rope makes an angle of 45° with respect to the vertical axis as shown in the figure. The magnitude of F is: [2024]

Q 2 :

A body of mass 1 kg is suspended with the help of two strings making angles as shown in figure. Magnitude of tensions $T_{1}$ and $T_{2}$ , respectively, are (in N): [2025]

Q 3 :

A body of mass m is suspended by two strings making angles $θ_{1}$ and $θ_{2}$ with the horizontal ceiling with tensions $T_{1}$ and $T_{2}$ simultaneously. $T_{1}$ and $T_{2}$ are related by $T_{1} = \sqrt{3} T_{2}$ . The angle $θ_{1}$ and $θ_{2}$ are [2025]

Q 4 :

A block of $\sqrt{3}$ kg is attached to a string whose other end is attached to the wall. An unknown force F is applied so that the string makes an angle of 30° with the wall. The tension T is (Given g = 10 ${ms}^{- 2}$ ) [2023]

Q 5 :

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 $30 °$ . Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is ________. [2026]

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