A 0.60 kg ladle sliding on a horizontal frictionless surface is attached to one end of a horizontal springPhy(k = 430 N/m) whose other end is fixed. The ladle has a kinetic energy of 8.6 J as it passes through its equilibrium position (the point at which the spring force is zero). (a) At what rate is the spring doing work on the ladle as the ladle passes through its equilibrium position? (b) At what rate is the spring doing work on the ladle when the spring is compressed 0.17 m and the ladle is moving away from the equilibrium position?

To determine the rate at which the spring is doing work on the ladle, we need to calculate the spring force and the velocity of the ladle at the given positions.

(a) At the equilibrium position, the ladle has a kinetic energy of 8.6 J. Since the spring force is zero at this position, the total mechanical energy (kinetic energy + potential energy) of the system is conserved.

At the equilibrium position:
Total mechanical energy = Kinetic energy + Potential energy
Total mechanical energy = 8.6 J + 0 J (since the potential energy is zero at equilibrium)
Total mechanical energy = 8.6 J

We can equate this to the potential energy stored in the spring at maximum compression:

Potential energy = (1/2)kx²

Where:
k = spring constant = 430 N/m
x = maximum compression = 0.17 m

Substituting the values:
8.6 J = (1/2)(430 N/m)(0.17 m)²
8.6 J = (1/2)(430 N/m)(0.0289 m²)
8.6 J = 0.05645 Nm

Therefore, at the equilibrium position, the spring is doing work on the ladle at a rate of 0.05645 Nm.

(b) When the ladle is compressed by 0.17 m and moving away from the equilibrium position, the potential energy stored in the spring is given by the same equation:

Potential energy = (1/2)kx²

Substituting the values:
Potential energy = (1/2)(430 N/m)(0.17 m)²
Potential energy = (1/2)(430 N/m)(0.0289 m²)
Potential energy = 0.05645 Nm

Since the ladle is moving away from the equilibrium position, its kinetic energy will be reducing. The difference in kinetic energy at different positions will give us the work done by the spring.

Work done by the spring = Initial kinetic energy - Final kinetic energy

The change in kinetic energy can be calculated using the work-energy theorem:

Work done by the spring = Change in kinetic energy

Therefore:
Work done by the spring = 8.6 J - 0 J (since the ladle is at rest)

So, at this position, the spring is not doing any work on the ladle.

In summary:
(a) At the equilibrium position, the spring is doing work on the ladle at a rate of 0.05645 Nm.
(b) When the spring is compressed 0.17 m and the ladle is moving away from the equilibrium position, the spring is not doing any work on the ladle.