A 2.36 kg block rests on a frictionless surface and is attached to an ideal spring with a spring constant k = 260 N/m. A force is applied to the block, moving it from an initial position xi = 5.89 cm to a final position xf = -15.4 cm, with each distance measured relative to the block's equilibrium position. Find (a) the work done by the spring and (b) the work done by the applied force.

Let's start by calculating the work done by the spring. We use the fact that the work done by or on a spring can be calculated using the formula:

Work by the spring (W_spring) = 1/2 * k * (x_initial^2 - x_final^2)

where k = spring constant (260 N/m), x_initial = initial displacement from equilibrium (5.89 cm = 0.0589 m), x_final = final displacement from equilibrium (-15.4 cm = -0.154 m).

a) To calculate the work done by the spring (W_spring), we first need to convert the displacements into meters as follows:

W_spring = 1/2 * 260 N/m * (0.0589 m^2 - (-0.154 m)^2) W_spring = 0.5 * 260 N/m * (0.00347041 m^2 - 0.023716 m^2) W_spring = 130 N/m * (-0.02024559 m^2) W_spring = -2.6319357 J

The work done by the spring is negative, which indicates that the spring force is doing negative work on the block. This corresponds to the spring force acting in the opposite direction of the displacement, which in this case, is pulling the block back towards the equilibrium point.

Now let's find the work done by the applied force (W_applied):

b) The work done by the applied force must be equal in magnitude and opposite in sign to the work done by the spring since there is no other force doing work on the system (we neglect gravity and any other potential forces since the surface is frictionless and we're not told of any other interactions). The total energy of the system is conserved as there is no non-conservative work done (like friction or air resistance). Hence:

W_applied = -W_spring W_applied = 2.6319357 J

We get that the work done by the applied force is positive, which means that the applied force did work to move the block from its initial to the final position, working against the force from the spring.