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Question

(A) When the body of mass $M$ is displaced by a small distance $x$ along the inclined plane,both springs are deformed. One spring is compressed by $x$

Vedclass · Accepted Answer

$2\pi \sqrt{\frac{M}{2K}}$

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(A) When the body of mass $M$ is displaced by a small distance $x$ along the inclined plane,both springs are deformed. One spring is compressed by $x$ and the other is stretched by $x$.The restoring force exerted by each spring is $F = -Kx$.Since both springs act in the same direction to restore the equilibrium position,the total restoring force is $F_{net} = -Kx - Kx = -2Kx$.The effective spring constant of the system is $K_{eff} = 2K$.The period of oscillation $T$ for a spring-mass system is given by $T = 2\pi \sqrt{\frac{M}{K_{eff}}}$.Substituting $K_{eff} = 2K$,we get $T = 2\pi \sqrt{\frac{M}{2K}}$.The inclination of the plane does not affect the period of oscillation because it only changes the equilibrium position of the mass,not the restoring force constant.

On a smooth inclined plane,a body of mass $M$ is attached between two springs. The other ends of the springs are fixed to firm supports. If each spring has a force constant $K$,the period of oscillation of the body (assuming the springs are massless) is

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