Question

In a system two weightless rigid bars AB and BC of length 'a' each having hinge supports at the' ends A and C, respectively, are connected to each other at B by a frictionless hinge (internal hinge). The rotation at the hinge is restrained by a rotational spring of stiffness k and system assumes a straight line configuration ABC. The rotation at the supports due to vertical load P acting at joint B is

• A. $\frac{Pa}{2k}$
• B. $\frac{Pa}{4k}$
• C. $\frac{Pa}{k}$
• D. $\frac{2Pa}{k}$

Answer 1

The correct option is B $\frac{Pa}{4k}$

Rotation in spring is double than that of rotation at the ends .



Strain energy stored in spring,

$U=\frac{1}{2}\times T\times 2\theta [\because T=k2\theta ]$

$=2k{\theta }^2$

Work done nby load,

$W=\frac{1}{2}\times P\times \delta$

But from the figure above we have $\theta =\frac{\delta }{a}$

Also, strain energy stored in the spring is equal to the work done by the load.

$\therefore U=W$

$2k{\theta }^2=\frac{1}{2}\times P\times \delta$

$\Rightarrow 2k{\theta }^2=\frac{1}{2}\times P\times a\theta$

$\Rightarrow \theta =\frac{Pa}{4k}$