Two charges, each equals to $$q,$$ are kept at $$x=-a$$ and $$x=a$$ on the $$x$$-axis. A particle of mass $$m$$ and charge $${q_0} = {q \over 2}$$ is placed at the origin. If charge $${q_0}$$ is given a small displacement $$\left( {y < < a} \right)$$ along the $$y$$-axis, the net force acting on the particle is proportional to

A sonometer wire of length $$1.5$$ $$m$$ is made of steel. The tension in it produces an elastic strain of $$1\% $$. What is the fundamental frequency of steel if density and elasticity of steel are $$7.7 \times {10^3}\,kg/{m^3}$$ and $$2.2 \times {10^{11}}\,N/{m^2}$$ respectively ?

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $$M.$$ The piston and the cylinder have equal cross sectional area $$A$$. When the piston is in equilibrium, the volume of the gas is $${V_0}$$ and its pressure is $${P_0}.$$ The piston is slightly displaced from the equilibrium position and released,. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frquency

The above $$p$$-$$v$$ diagram represents the thermodynamic cycle of an engine, operating with an ideal monatomic gas. The amount of heat, extracted from the source in a single cycle is