Final answer:
The energy of a mechanical wave on a string increases by a factor of four if the amplitude is doubled, even when the linear density of the string is halved.
Step-by-step explanation:
The question asks what happens to the energy of a mechanical wave on a string if the amplitude is doubled and the linear density of the string is halved. In the realm of physics, particularly when discussing waves, the energy carried by a mechanical wave is proportional to the square of its amplitude. This means that if the amplitude is doubled, the energy carried by the wave is increased by a factor of four (quadrupled).
On the other hand, the halving of the linear density of the string affects the speed of the wave, but the question doesn't provide sufficient information to determine the effect on energy directly from this change alone. However, assuming tension remains constant and only linear density is reduced, the speed of the wave would increase but that does not influence energy in the same way amplitude does. Thus, accounting only for the change in amplitude, the energy of the wave increases by a factor of four (c).