Understanding Vertical Mounted Spring Calculations with Conservation of Energy

[BlueCardBird]

Can anyone briefly explain to me when using conservation of energy to calculate for vertically mounted spring questions, why gravitational potential energy is neglected?

 

[haruspex]

Do you mean as it pertains to the mass of the spring? Usually you are told the spring is light and some other mass is affected by both spring and gravity. Sure, you could allow for the spring's weight too, and probably would in a real situation, but it usually doesn't add anything very interesting to the algebra.

 

[cosmic dust]

Can anyone briefly explain to me when using conservation of energy to calculate for vertically mounted spring questions, why gravitational potential energy is neglected?

Because the gravity field just changes the height that the hanging mass balances. So gravity's effective action is just to expand the natural length of the spring. Now, because the gravity potential is linear, equal changes in height result in the same change in gravity potential, regardless of the height those changes happen. To clarify the above, take a look at these equations about the equation of motion and the total potential energy in 2 cases: a) in absence of gravity and b) with gravity. We assume that z = 0 is the point where the hanging mass experiences no spring force in the absence of gravity.

a) F = -k z
V = $\frac{1}{2}$ k z²

b) F = -k z - m g = -k (z+z₀) , where z₀ = mg/k
you see that the mass balances now at z = -z₀ , a little longer than before. So you can study the problem using the new variable ζ = z + z₀ . The equation of motion will be:
F = -k ζ
which is equivalent to a) case.


V = $\frac{1}{2}$ k z² + m g z = $\frac{1}{2}$ k (z+z₀)² - $\frac{1}{2}$ k z₀²

Since constant terms in potential energy have no physical significance, you can drop them and define the equivalent potential function:
V^* = $\frac{1}{2}$ k ζ²

Compare the equation of motion and the potential function in a) and b) cases, and get your answer!

 

[sophiecentaur]

The mass of the spring will, however, have an effect on the period of oscillation because it's part of the oscillating system.

 

[Chestermiller]

If the mass of the spring is regarded as a distributed parameter along its length, then including the spring mass can add significant complexity to the analysis. Sometimes what people do to take this into account is to approximate the spring as a sequence of masses joined by massless springs.

 

[nasu]

Can anyone briefly explain to me when using conservation of energy to calculate for vertically mounted spring questions, why gravitational potential energy is neglected?

The way you formulate it, it is symply not true. In some problems you do not neglect the gravitational PE.
If you are talking about harmonic motion, maybe. If you are looking at a problem like a bloc falls from 2 m on top of a spring, then no, you don't neglect it.