A block, initially at rest, has a mass m and sits on aplane inclined at angle (theta). It slides a distance d before hitting a spring and compresses the spring by a maximum distance of xf. If the coefficient of kinetic friction between the plane and block is uk, then what is the force constant of the spring?

Answer: k = ma + uk*mgcosθ / xf

Explanation: The body is placed on a frictionless inclined ramp.

The weight of the object has 2 components, horizontal component (mgsinθ) and vertical component (mgcosθ).

The horizontal component of weight is responsible for making tje object slide down the plane even with no applied force.

So from newton's second law of motion

mgsinθ - uk*R = ma

Where uk = coefficient of kinetic friction.

R = normal reaction = mgcosθ

mgsinθ - uk*mgcosθ = ma

mgsinθ = ma + uk*mgcosθ

mgsinθ is the applied force in this case. This applied force compresses a spring.

According to hooke's law,

F = ke

Where F = ma + uk*mgcosθ, e = xf

F = applied force, e = extension and k = spring constant.

k = F/e

k = ma + uk*mgcosθ / xf