Question

8. A mass on the end of a string of length is displaced an angle from vertical and released (shown in the figure below). At the bottom of the swing the mass is released from the string and allowed to slide with friction coefficient uk) over a distance, d, before compressing a spring a distance, x (you may assume the friction only acts over the distance, d). CHEMISTRO IS BACK Use conservation of energy to find an equation for the distance the spring is compressed, x, as a function of the mass, the length of the string, the initial angle of displacement, the coefficient of kinetic friction and the spring constant, k. Fo000 Extra Credit: Solve problem 8 assuming the more realistic case that the friction acts over distance x as well. Simplify the answer as much as possible for full credit. (5 points)

Answer 1

By conservation of energy,

Initial gravitational potential energy = final spring potential energy +energy dissipated due to friction

mgL*(1 - cos theta) =0.5kx^2 + uk mgd

x = sqrt(2(mgL*(1 - cos theta) -u_k mgd)/k)

Extra credit: Again by energy conservation,

Initial gravitational potential energy = final spring potential energy +energy dissipated due to friction

mgL*(1 - cos theta) =0.5kx^2 + uk mg(d+x)

Let mgL*(1 - cos theta)=A, uk mg=B,

we have A = 0.5kx^2 + B(d+x)

2 Ak+B2 - 2 B dk - B and k0

x = [sqrt(2k* mgL*(1 - cos theta) +(u_k mg)^2 -2*u_k mg*d*k) -u_k mg]/k