Question

A block of mass M is moving with a certain velocity on a horizontal surface. It approaches the free end of a spring shown in the figure with a velocity u. The other end of the spring (assumed to be ideal) with force constant k is fixed to a rigid, vertical wall.

(i) Calculate the maximum compression produced in the spring if the horizontal surface is smooth. 

(ii) Let the horizontal surface shown above be increasingly rough towards the spring and the coefficient of friction between the block and the surface varies as μ = ax where a is a constant with appropriate dimensions and x is the instantaneous compression in the spring. What will be the maximum compression produced in the spring in this case?

Answer 1

Case (i) 

When the surface is smooth; the entire KE of the block is stored in the spring as elastic PE. If x₀ is the maximum compression produced in the spring; we have

When the surface is rough; work has to be done to overcome friction and to cause a compression in the spring. 

For a compression ‘x’ in the spring, we have μ = ax .

∴ Force of friction for a compression x is

Work done against friction for a maximum compression x₁ (say) is

Energy stored in the spring = 1/2 K \(x^2_1\)

From law of conservation of energy;

Answer 2

The energy stored in the spring exists in the form of the potential energy and it is given as:

PE=1/2 kx^2

When the block is left from maximum compression, the potential energy gets converted into kinetic energy of the block.

KE=1/2 mv^2

At maximum compression
KE=PE
∴1/2 kx^2 = 1/2 mv^2
⇒X= root m/k .v
The magnitude of velocity will be the same but the direction will be opposite.