spring max displacement: please use conservation of energy method

Question

The figure below shows a 10-kg copper block connected to an ideal spring having

stiffness constant k = 130 N/m with the spring initially at its natural length. The copper

block is presently held in place and there is a constant wind with force F blowing to the

right as shown below.

The spring is initially at its natural length

If the copper block is now released,

a.) determine how strong the wind force must be if the copper block is to start

moving to the right along the steel tabletop.

b.) If the wind force is larger than your result computed in part (a), say F = 80 N,

determine the maximum amount that the spring will compressed after the copper block is

released.

c.) If at this moment, the wind were to stop blowing (so that F = 0), determine the

maximum amount that the spring will stretch.

Answer 1

There are lots of conceptual ways you can do this problem with the conservation of energy. I will focus on not moving the equilibrium point before and after adding the force.

a) Any force will start to move the block to the right if the block starts at its rest length.

b) The energy at the start will be 0 J.

The work added to it will be

W = Fd.

The final energy will be

1/2 k x²

E = 0J + F x = 1/2 k x²

x = 2 F/k = 2 (80 N) / (130 N/m)

x = 16/13 m

c) The maximum amount of displacement occurs when all of the energy is potential and the box briefly stops. The total energy added was W = F d

E = F d = 2 F² / k= 1/2 k x²

x = 2 F/x

x = 16/13 m

This make sense as it started at its maximum displacement. If you saw that you can also treat this as a conceptual question.