Physics. Elastic and Potential Enegry?

Question

A childs toy consists of a piece of plastic attached to a spring. The spring is compressed against the floor a distance of 2.00 cm, and the toy is released. If the toy has a mass of 100 g and rises to a maximum height of 60.0 cm, estimate the force constant of the spring

Answer 1

Potential energy
Pe= mgh h= H-x
Potential energy of the spring
Pes= kx^2
Potential energy of the spring Pes gets converted to max kinetic energy Ke (at x =2 cm) and Ke gets converted to potential energy Pe (at x=60cm). Finally
Pe=Pes
mg(H-x) = kx^2
k= mg(H-x)/x^2
k= .100 x 9.81 ( 0.60 - 2)/ (0.02)^2
k=1420 N/m

Answer 2

PE(H) = mgH = k/2 delX^2 = PE(S); where mgH is the potential energy of mass m = .1 kg when at height H = .6 m above the compressed spring. k = the spring coefficient you are looking for; delX = .02 m the compression. PE(S) = F/2 delX = k/2 delX delX is the potential energy of the compressed spring.

F/2 is the average force used to compress the spring delX. We do not use F, because that is the max F when the spring is fully compressed (F = k delX). As F is linear wrt delX, its average over delX is F/2.

PE(H) = PE(S) because of the conservation of energy. That is potential energy in the spring is converted to potential energy at max height H where there is no kinetic energy. Thus, from mgH = k/2 delX^2, we have k = 2mgH/delX^2 and g = 9.81 m/sec^2 on Earth's surface. Everything on the RHS is known; so you can do the math.

Note H is measured from the tip of the compressed spring. As that is "against the floor," H is about the same to the floor of the launch site.

Also note, we don't care what the kinetic energy might be off the spring. That's because the conservation of energy says the potential energy of the spring has to equal the potential energy at max height regardless of what the kinetic energy might have been. And that results because KE(H) = 0 when all the potential energy in the spring is once again potential energy, but this time due to its height H.