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blalien
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[SOLVED] Mass on a spring
This problem is from Gregory’s Classical Mechanics
A light spring of natural length a is placed on a horizontal floor in the upright position. When a block of mass M is resting in equilibrium on top of the spring, the compression of the spring is a/15. The block is now lifted to a height 3a/2 above the floor and released from rest. Find the compression of the spring when the block first comes to rest.
Mgh = the potential energy of the mass at a height h above the floor
1/2kx^2 = the potential energy of the spring, since we are assuming that the spring obeys Hooke's Law
It doesn’t help that the book never defines “natural length” or “compression.” But anyway, this is my logic:
Assume that, on the horizontal floor, V = 0. So, you have two systems and two equations:
Mass starts at rest on spring -> Mass compresses spring by a/15
Mga = Mg(14a/15)+1/2k(a/15)^2
Mass starts at 3a/2 above floor -> Mass compresses spring by x
Mg(3a/2) = Mg(a-x)+1/2kx^2
Then you solve for k and x. Unfortunately, this yields the wrong answer, which is apparently x = a/3. I don’t want to be told exactly how to do the problem. But, could you please just point to the faulty part in my logic? I would really appreciate the help.
Thanks!
Homework Statement
This problem is from Gregory’s Classical Mechanics
A light spring of natural length a is placed on a horizontal floor in the upright position. When a block of mass M is resting in equilibrium on top of the spring, the compression of the spring is a/15. The block is now lifted to a height 3a/2 above the floor and released from rest. Find the compression of the spring when the block first comes to rest.
Homework Equations
Mgh = the potential energy of the mass at a height h above the floor
1/2kx^2 = the potential energy of the spring, since we are assuming that the spring obeys Hooke's Law
The Attempt at a Solution
It doesn’t help that the book never defines “natural length” or “compression.” But anyway, this is my logic:
Assume that, on the horizontal floor, V = 0. So, you have two systems and two equations:
Mass starts at rest on spring -> Mass compresses spring by a/15
Mga = Mg(14a/15)+1/2k(a/15)^2
Mass starts at 3a/2 above floor -> Mass compresses spring by x
Mg(3a/2) = Mg(a-x)+1/2kx^2
Then you solve for k and x. Unfortunately, this yields the wrong answer, which is apparently x = a/3. I don’t want to be told exactly how to do the problem. But, could you please just point to the faulty part in my logic? I would really appreciate the help.
Thanks!