Maximum Compression for a Block on Spring

[Abu]

Homework Statement

A spring of negligible mass has force constant k = 800 N/m. You place the spring vertically with one end on the floor. You then lay a 1.6 kg block on top of the spring and release the book from rest. Find the maximum distance the spring will be compressed

Relevant Equations

F = kx

Hi everyone, just a quick question..

I tried this problem using Newtons laws, not conservation of energy, and I got an answer exactly half of what the correct answer is, and I'm not sure why. Here is what I did:

Net force = zero once the spring is compressed, therefore

mg - kx = 0
mg = kx
mg/k = x
1.6(9.8)/800 = x
x = 1.96 cm

The actual answer is 3.92 cm however, and I don't know why.

If I do this problem using conservation of energy, I get the correct answer but I can't explain why my Newtons laws method is wrong.

Thanks for your consideration!

 

[Antarres]

When you let go of the block and it starts compressing the spring, the spring will start oscillating. The equation you were using is at the equilibrium position where gravity and elastic force cancel. However, once block reaches this position, it will have non-zero velocity, so it will move past this point and compress the spring further, until it's velocity drops to zero. At this point, elastic force will be bigger than gravity, so it will push the block up, hence it will oscillate.

Therefore, since the energy is conserved, in order to get the maximum compression, you need the equation where kinetic energy is zero, so by substituting correct potential energy for gravity and elastic force, you get the right answer, as you did. In order to get that same point via Newton's law, you'd need to search for the point where elastic force is maximal, not the point where it cancels with gravity. This would prove more complicated than using energy conservation in my opinion, so that's why this exercise is most likely intended to be done using energy conservation.

Hope that helps.

 

[Abu]

When you let go of the block and it starts compressing the spring, the spring will start oscillating. The equation you were using is at the equilibrium position where gravity and elastic force cancel. However, once block reaches this position, it will have non-zero velocity, so it will move past this point and compress the spring further, until it's velocity drops to zero. At this point, elastic force will be bigger than gravity, so it will push the block up, hence it will oscillate.

Therefore, since the energy is conserved, in order to get the maximum compression, you need the equation where kinetic energy is zero, so by substituting correct potential energy for gravity and elastic force, you get the right answer, as you did. In order to get that same point via Newton's law, you'd need to search for the point where elastic force is maximal, not the point where it cancels with gravity. This would prove more complicated than using energy conservation in my opinion, so that's why this exercise is most likely intended to be done using energy conservation.

Hope that helps.

Thank you so much for your fast reply. Your response helped me a lot, I really appreciate it! Thanks!

 

[Antarres]

You're welcome!