question_answer

A chord ACB, 5m long is attached at points A and B to the vertical walls 3 m apart.

A pulley of negligible mass and negligible radius carries 200 N load is free to roll over chord without friction. Dimension x in figure, when pulley is in equilibrium is

A) \[\frac{9}{8}m\]                        

B)  \[\frac{7}{5}m\]

C) \[\frac{4}{3}m\]                        

D) \[\frac{3}{4}m\]

Correct Answer: A

Solution :

Geometry of given arrangement is as shown below.

Triangles\[\Delta ADC\], \[\Delta BEC\]and \[\Delta BGF\] are similar triangles. Also, \[\Delta ACD\]is congruent to\[\Delta FCD\]. So, AC = FC.

Hence,\[FB=5\text{ }m\]

In \[\Delta BGF\],

\[\cos \theta =\frac{base}{hypotenuse}=\frac{3}{5}\]

So, \[\sin \theta =\frac{4}{5}and\tan \theta =\frac{4}{3}\]

Now, \[\Delta AMC\cong \Delta NMC\]     (by AAS rule)

AM=MN

Also, \[AM+MN+ND=3m\]

\[\Rightarrow NP=3-2x\left( \therefore AM=x\,and\,MN=x \right)\]

Now in \[\Delta BPN\]

\[\tan \theta =\frac{BP}{NP}\Rightarrow \frac{4}{3}=\frac{1}{3-2x}\Rightarrow x=\frac{9}{8}m\]