Energy conservation with conservative forces: A box of mass m is pressed against (but is not attached to) an ideal spring of force constant k and negligible mass, compressing the spring a distance x. After it is released, the box slides up a frictionless incline as shown in the figure and eventually stops. If we repeat this experiment but instead use a spring having force constant 2k

Question

Quizplus · Accepted Answer

Conservation of energy applies in this system since the only forces acting on the box are conservative (gravity and the spring force). Therefore, we can equate the initial potential energy of the spring (at compression distance x) to the final potential energy of the box at its maximum height on the incline.

Initial potential energy: 
U_initial = 1/2 k x^2 
Final potential energy: 
U_final = mgh

Since the incline is frictionless, there is no work done by non-conservative forces and we have the conservation of mechanical energy:

U_initial = U_final + 1/2 mv^2

where v is the velocity of the box at the instant it loses contact with the spring.

Now we can analyze the different scenarios:

A) If we use a spring with force constant 2k, the initial potential energy will be larger since U_initial = 1/2 (2k) x^2 = 2U_initial. Therefore, the final potential energy (at the maximum height) will also be doubled, and the box will go up twice as high.

B) If the initial potential energy is doubled, the total mechanical energy of the system will also be doubled. Therefore, just as the box moves free of the spring, the kinetic energy of the box will be twice as great as before.

C) The kinetic energy of the box can also be related to the speed v:

1/2 mv^2 = U_initial - U_final = 1/2 kx^2 - mgh

If we double the force constant k in the spring, the compression distance x will be reduced (since we are applying the same force to the box), and therefore, the initial potential energy will be lower. However, this will be compensated by the larger force constant, resulting in equal kinetic energies:

1/2 mv^2 = 1/2 (2k)x^2 - mgh = kx^2 - mgh

1/2 mv^2 = 1/2 kx^2 - mgh

Therefore, the speed of the box just as it moves free of the spring will be the same in both cases.

Thus, all three choices are correct and the answer is D.

Energy Conservation with Conservative Forces: a Box of Mass M