Blocks on Spring without Friction

AI Thread Summary
The discussion centers on calculating the potential and mechanical energy of a block attached to a spring on a frictionless surface. The potential energy of the block when the spring is stretched is incorrectly calculated using the formula PE = mgh, which is not applicable in this context. The total mechanical energy of the block remains constant throughout its motion, as energy is conserved in a frictionless environment. When the spring returns to its unstretched position, the block's speed can be determined using energy conservation principles. The calculations for the block's speed at various positions, including halfway back to the unstretched position, are also explored.
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A spring is stretched a distance of Dx = 40 cm beyond its relaxed length of xo by a force, F = 20 N. Attached to the end of the spring is an block of mass m = 12 kg, which rests on a horizontal frictionless surface.
After a moment, the force is removed.

a) What is the potential energy of the block due to the spring when it is held in the stretched position?

For this question, I did PE=(mgh). So, PE= (12)(9.8)(0.4m) and got 47.04 J, but that's not the right answer.

b) What is the total mechanical energy of the block when it is held in the stretched position?

c) What is the total mechanical energy of the block when it returns to its unstretched position?

d) When the spring again returns to its unstretched length, what is the speed of the block?

e) What is the total mechanical energy of the block when it has returned only halfway (20 cm) to its unstretched position?

f) When the spring has returned only halfway (20 cm), what is the speed of the block?
 
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