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 compress the spring a distance of 2x

Question

Quizplus · Accepted Answer

From energy conservation, we know that the initial potential energy stored in the spring is equal to the final kinetic energy of the box.

When the spring is compressed a distance x, the potential energy stored in the spring is (1/2)kx^2. When the box slides up the incline, its potential energy is converted into kinetic energy until it eventually stops at maximum height h.

Thus, (1/2)kx^2 = (1/2)mv^2 + mgh

where v is the velocity of the box just as it moves free of the spring, and g is the acceleration due to gravity.

If we repeat the experiment but compress the spring a distance of 2x, the initial potential energy stored in the spring is (1/2)k(2x)^2 = 2kx^2, which is twice the initial potential energy in the first experiment.

Since the potential energy is now twice as large, the final kinetic energy of the box will also be twice as large. Thus, just as it moves free of the spring, the box will be traveling twice as fast as before. Therefore, choice B is correct.

Energy Conservation with Conservative Forces: a Box of Mass M