First slide

Mechanical equilibrium force cases

Question

A body of mass √3 kg is suspended by a string to a rigid support. The body is pulled horizontally by a force F until the string makes an angle of 30° with the vertical. The value of F and tension in the string are.

Moderate

A

9.8 N, 9.8 N
B

9.8 N, 19.6 N
C

19.6 N, 19.6 N
D

19.6 N, 9.8 N

Solution

The body is in equilibrium under the action of the forces
$\vec{F_{g}}, \vec{F} and \vec{T}$ as shown in figure.
By Lami's Theorem,

$\large \frac{F}{{\sin ({{180}^0} - \theta )}} = \frac{T}{{\sin {{90}^0}}} = \frac{{{F_g}}}{{\sin ({{90}^0} + \theta )}}$

$\large \therefore F = \frac{{{F_g}\sin \theta }}{{\cos \theta }} = mg\tan \theta = \sqrt 3 \times 9.8 \times \tan {30^0} = 9.8N$

and
$\large T = {F_g} \times \frac{1}{{\cos \theta }} = \frac{{mg}}{{\cos \theta }} = \frac{{\sqrt 3 \times 9.8}}{{\frac{{\sqrt 3 }}{2}}} = 19.6N$

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