Harmonic oscillator with/without gravity

[daudaudaudau]

If I have mass on a spring that is oscillating in a linear motion, this system has a certain energy. Now if we imagine the system to be aligned along the vertical, why is the energy lower when gravity is turned on? I can calculate it and see that it is correct, but what is the "explanation" ? Because, I mean, when gravity is turned on, the string is stretched further, so this should increase the energy.

 

[Doc Al]

When the system is vertical, you must also consider gravitational PE along with spring PE. Compare the increase in spring energy (as it stretches to the new equilibrium point, say) with the decrease in gravitational PE.

 

[daudaudaudau]

Right, and I can see that the decrease in gravitational PE is larger than the increase in the spring PE. Is there some general "law" stating that whenever you apply an external conservative force to a system, the energy decreases?

 

[Doc Al]

Right, and I can see that the decrease in gravitational PE is larger than the increase in the spring PE.

That's only true if you compare the change in gravitational and spring PE as the masses lowers to its new equilibrium position. Since the gravitational force is constant, the spring force will soon overtake it. I don't see anything particularly significant about this. With respect to its new equilibrium point, a vertical spring+mass behaves similarly to a horizontal one.

Is there some general "law" stating that whenever you apply an external conservative force to a system, the energy decreases?

Not that I'm aware of.

 

[PJC01]

The addition of gravity to a harmonic oscillator system does indeed result in a decrease in energy. This can be explained by considering the forces acting on the mass on the spring. In a simple harmonic oscillator without gravity, the only force acting on the mass is the restoring force of the spring, which is directly proportional to the displacement of the mass from its equilibrium position. This results in a sinusoidal motion with a constant amplitude and energy.

However, when gravity is turned on, an additional force is introduced. Gravity pulls the mass downwards, causing it to stretch the spring further and increase the potential energy of the system. This results in a larger amplitude of oscillation and therefore a higher potential energy. However, as the mass moves downwards, it also gains kinetic energy, which allows it to overcome the increased potential energy and continue oscillating.

As the mass moves upwards, the opposite occurs. The potential energy decreases, but the kinetic energy also decreases as the mass slows down due to the restoring force of the spring. This results in a decrease in the total energy of the system.

In summary, the addition of gravity changes the balance between potential and kinetic energy in the system. This results in a decrease in the total energy of the system compared to a harmonic oscillator without gravity.