Two blocks are connected by a light string that passes over two frictionless pulleys as in the figure below. The block of mass m2 is attached to a spring of force constant k and m1 > m2. If the system is released from rest, and the spring is initially not stretched or compressed, find an expression for the maximum displacement d of m2. (Use any variable or symbol stated above along with the following as necessary: g.)d =
Question
Two blocks are connected by a light string that passes over two frictionless pulleys as in the figure below. The block of mass m2 is attached to a spring of force constant k and m1 > m2. If the system is released from rest, and the spring is initially not stretched or compressed, find an expression for the maximum displacement d of m2. (Use any variable or symbol stated above along with the following as necessary: g.)d =
Solution
The problem involves the conservation of energy. Initially, when the system is at rest, all the energy in the system is potential energy. When the block of mass m1 falls and the block of mass m2 rises, the potential energy is converted into kinetic energy. At the maximum displacement d of m2, the system momentarily comes to rest, so all the energy is once again potential energy.
Let's denote the initial height of m1 as h1 and the final height of m1 as h2. The initial height of m2 is h2 and the final height of m2 is h1 + d. The potential energy of m1 initially is m1gh1 and finally is m1gh2. The potential energy of m2 initially is m2gh2 and finally is m2g(h1 + d). The potential energy stored in the spring at maximum displacement is (1/2)kd^2.
According to the conservation of energy, the total energy initially (potential energy) should be equal to the total energy finally (potential energy + spring energy). Therefore, we can write the equation as:
m1gh1 + m2gh2 = m1gh2 + m2g(h1 + d) + (1/2)kd^2
Solving this equation for d will give us the maximum displacement of m2.
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