Solveeit Logo
Login

Question

Question: A uniform beam of length 2a rests in equilibrium against a smooth vertical plane and over a smooth p...

A uniform beam of length 2a rests in equilibrium against a smooth vertical plane and over a smooth peg at a distance h from the plane. If q be the inclination of the beam to the vertical, then sin3q is

A

ha\frac{h}{a}

B

h2a2\frac{h^{2}}{a^{2}}

C

ah\frac{a}{h}

D

a2h2\frac{a^{2}}{h^{2}}

Answer

ha\frac{h}{a}

Explanation

Solution

Let AB be a rod of length 2a and weight W. It rests against a smooth vertical wall at A and over peg C, at a distance h from the wall. The rod is in equilibrium under the following forces :

(i) The weight W at G

(ii) The reaction R at A

(iii) The reaction S at C perpendicular to AB.

Since the rod is in equilibrium. So, the three force are concurrent at O.

In DACK, we have, sin q =hAC\frac{h}{AC}

In DACO, we have, sin q = AOa\frac{AO}{a}

In DAGO, we have sin q = AOa\frac{AO}{a};

∴sin⁡3θ=hAC.ACAO.AOa=ha\therefore\sin^{3}\theta = \frac{h}{AC}.\frac{AC}{AO}.\frac{AO}{a} = \frac{h}{a}