A spring of spring constant k placed horizontally on a rough horizontal surface is compressed against a block of mass m placed on the surface so as to store maximum energy in the spring.If the coefficient of friction between the block and the surface is μ, find the potential energy stored in the spring.
Answers
The greater the compression, the greater will be restoring force of the spring as restoring force of the spring is given by ;
F = -k x .........(1)
where, k is the spring constant and x is the compression produced in the spring.
As the spring is compressed toward left, the restoring force will act in right direction. Due to this, the block of mass m attached to the spring will move towards right and the frictional force will act directed opposite to the movement of the block.
Given , coefficient of friction between the block and the surface is μ, As, μ = F/R = F/mgμ, As, μ = F/R = F/mg
F = μ mg ..........(2)μ mg ..........(2)
where, F is the coefficient of kinetic friction and R is the normal reaction.
In equilibrium position, the spring along with mass m, will return to its mean position.
In this condition,
eq. 1 = eq. 2
i.e., -k x = μμ mg
or x = − μ m gk- μ m gk
Since, potential energy of the spring is given by the relation :
P.E. = 1/2 k x2
= 12 k (−μ mgk)2= 12(μ2 m2 g2k )12 k -μ mgk2= 12μ2 m2 g2k