Two blocks are connected by a light string that passes over two frictionless pulleys. 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.

(BELOW YOU CAN FIND ATTACHED THE IMAGE OF THE SITUATION)

Explanation:

For this we're going to use conservation of mechanical energy because there are nor dissipative forces as friction. So, the change on mechanical energy (E) should be zero, that means:

(1)

With the initial kinetic energy, the initial potential energy, the final kinetic energy and the final potential energy. Note that initialy the masses are at rest so , when they are released the block 2 moves downward because m2>m1 and finally when the mass 2 reaches its maximum displacement the blocks will be instantly at rest so . So, equation (1) becomes:

(2)

At initial moment all the potential energy is gravitational because the spring is not stretched so and at final moment we have potential gravitational energy and potential elastic energy so , using this on (2)

(3)

Additional if we define the cero of potential gravitational energy as sketched on the figure below (See image attached), and we have by (3) :

(4)

Now when the block 1 moves a distance d upward the block 2 moves downward a distance d too (to maintain a constant length of the rope) and the spring stretches a distance d, so (4) is:

dividing both sides by d

, with k the constant of the spring and g the gravitational acceleration.

at positions a, g kinetic energy is maximum.

at position d, potential energy is maximum.