Solving for Block Position After 3.3J Work on Spring-Block System

[mbrmbrg]

In the figure below, we must apply a force of magnitude 89 N to hold the block stationary at x = -2.0 cm. From that position we then slowly move the block so that our force does +3.3 J of work on the spring-block system; the block is then again stationary. What is the block's position? (There are two answers.)

The figure is a block resting on a horizontal surface, attatched to a horizontal spring.

-our force=spring force=-kd
so -89J=-k(-0.02m)==>k=-4450N/m

I then said that $$W=-(k\frac{x_f^2}{2}-k\frac{x_i^2}{2})$$, subbed in W=3.3 J, k=-4450N/m, x_i=-0.02m, and got the right answer (which is +/- 4.3cm).
But why did that equation work? 3.3 J is our work, not the spring's?

 

[PhanthomJay]

In the figure below, we must apply a force of magnitude 89 N to hold the block stationary at x = -2.0 cm. From that position we then slowly move the block so that our force does +3.3 J of work on the spring-block system; the block is then again stationary. What is the block's position? (There are two answers.)

The figure is a block resting on a horizontal surface, attatched to a horizontal spring.

-our force=spring force=-kd
so -89J=-k(-0.02m)==>k=-4450N/m

I then said that $$W=-(k\frac{x_f^2}{2}-k\frac{x_i^2}{2})$$, subbed in W=3.3 J, k=-4450N/m, x_i=-0.02m, and got the right answer (which is +/- 4.3cm).
But why did that equation work? 3.3 J is our work, not the spring's?

Yes, that is the work done by the person on the spring, The work results in stored elastic potential energy in the spring. If you look at the work energy theorem
$$W_{net} = \Delta K.E.$$ and since there is no KE change,
$$W_{net} = 0$$. But the net work is the sum of the work by the conservative forces (spring force) and non conservative forces (the 'we' pushing force). They must therefore be equal and opposite. We do so many joules of work, and the spring force does the same in the other direction.
Alternatively, you can use the total work energy method
$$KE_i + PE_{gravity}_i +PE_{spring}_i = KE_f + PE_{gravity}_f +PE_{spring}_f + W_{n.c.}$$
and since there is no kinetic or gravitational potential energy change, you get the same result for the work done by "we".

 

[kyle8921]

The equation you used, W=-(k\frac{x_f^2}{2}-k\frac{x_i^2}{2}), is known as the work-energy theorem. It states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the external force (89 N) is equal to the change in the potential energy of the spring-block system. This change in potential energy is due to the displacement of the block from its initial position (-0.02m) to its final position (x_f). By setting the work done by the external force equal to the change in potential energy, you can solve for the final position of the block. This equation works because energy is conserved in a closed system, and in this case, the only external force acting on the system is the external force applied to the block. Therefore, the work done by this external force is equal to the change in potential energy of the system.