Two blocks of mass m and 2m are initially motionless and connected to each other

Question

Two blocks of mass m and 2m are initially motionless and connected to each other by a massless string which slips frictionlessly over a pulley. The heavier mass starts at a distance H above the top of a massless spring with spring constant k.

a. When released, what is the downward acceleration of the heavier block in terms of m, H, k, and g, or a subset of those quantities?

b. Using the Work-Energy Theorem calculate the speed of the block when it first touches the top of the spring. Again express your answer in terms of m, H, k, and g, or a subset of those quantities. Important: You can check your answer by comparing to what your answer to part (a) would predict, but this part will only be graded on your use of the Work-Energy Theorem.

c. We are interested in knowing by what distance D the spring will be compressed when the heavier block comes to a stop. Again using the Work-Energy Theorem find an equation which, if solved, would tell you the value of D. You do not need to solve this equation. You should assume that the masses are pulled by the string with non-zero tension the entire time that the heavier mass is in motion. As before, express your answer in terms of m, H, k, and g, or a subset of those quantities.

Explanation / Answer

2m*g - T = 2m * a
T - m*g = m*a
(a)
Solving these eq,
2*mg - m*g = 2m*a + m*a
m*g = 3m*a
a = g/3
downward acceleration of the heavier block, a = g/3

(b)
Using Work Energy Theorem,
Initial Potential Energy + Initial Kinetic Eergy = Final Potential Energy + Final Kinetic Eergy
2m*g*H = 1/2*2m*v^2 + 1/2*m*v^2 + m*g*H
3/2*v^2 = g*H
v^2 = 2/3 *g*H
v = sqrt((2/3)*g*H)

(c)
Let the spring is compressed by Distance D.
Assuming that the lighter mass also stops when the heavier block comes to a stop.

Initial Kinetic Energy + Initial Potential Energy = Final Spring Potential Energy
1/2 * 2mv^2 +1/2*m*v^2 + 2m*g*D = 1/2*K*D^2 + m*g*D
3/2*m*v^2 = 1/2*k*D^2 - m*g*D
3/2 * m * 2/3 *g*H = 1/2*k*D^2 - m*g*D
m*g*H = 1/2*k*D^2 - m*g*D
1/2*k*D^2 - m*g*D - m*g*H = 0

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