Mass-spring system with friction

[GoSS190]

A mass M slides across a horizontal table. It collides with a spring, compresses the spring, and then the mass-spring system rebounds. This system can be used to find the spring constant k. When the mass first hits the spring at x = 0, it has speed v0.

a.) Let the coefficient of kinetic friction be μk. Assume that the spring plus mass compresses to a distance l, rebounds, and stops when it returns to x = 0, having compressed the spring only once. Use the work-energy relation, W_non-cons = ΔPE + ΔKE to find the required coefficient of friction μk in terms of l, v0 and g.

b.) Next, consider just half the cycle, with the mass starting out with speed v0 at position x = 0, and stopping (for an instant) at x = l. Use the work-energy relation once again to find a relation between k, l, v0, μk, and g.

c.) Use the results of parts (a) and (b) to solve for the spring constant k in terms of v0, μk, and g.

 

[LowlyPion]

How would you think to go about solving it other than putting it here for someone else to solve?

 

[CompuChip]

Yes, very interesting.
Any ideas? Formulas you might be able to use?

 

[GoSS190]

Well i know that KE is equal to 1/mv^2 and the potential energy of a spring is equal to 1/2kl^2. The work done by friction is going to be equal to μkmgl

 

[LowlyPion]

Won't the work done by friction be 2*μ_k*m*g*l ? (Twice the distance - to compression and back to equilibrium.)

So maybe add it all up and solve for μ_k?