2) A particle of mass m slides down a plane inclined at an angle , through a hei

Question

2) A particle of mass m slides down a plane inclined at an angle , through a height of H, as shown below. It then collides with a mass M sticking to it. Both masses then collide with a spring of spring constant k and you intercede to stop all motion exactly when the spring is at maximum compression. Assume that the coefficient of friction between the mass m and the inclined plane is and that this is the only friction in the problem, i.e., the horizontal surface is frictionless a) What is the total work done by friction on the system? b) How fast are both masses moving when they collide with the spring? c) What is the maximum compression of the spring?

Explanation / Answer

a] work done by friction= - umg cos theta L

= - umg cos theta H / sin theta

= -u mg H / tan theta

b] KE of mass m = mgH - umgH / tan theta

v = sqrt(2KE/m) = sqrt(2gH- 2ugH/tan theta)

now using momentum conservation, velocity of masses V= mv/(M+m)

= m*sqrt(2gH- 2ugH/tan theta) /(M+m) answer

c] maximum compression x is given by

0.5 kx^2 = 0.5 (M+m) V^2

x = sqrt((M+m)/k) V

= sqrt((M+m)/k) m*sqrt(2gH- 2ugH/tan theta) /(M+m) answer

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