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