A 10.0 kg block is released from point A in the figure below. The track is frictionless except for the portion between points B and C, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2300 N/m, and compresses the spring to 0.200 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between B and C.

To find the coefficient of kinetic friction between the block and the rough surface between points B and C, we can follow these steps:

1. Determine the gravitational force acting on the block.
- The gravitational force can be calculated using the formula: F_gravity = m * g, where m is the mass of the block and g is the acceleration due to gravity (9.8 m/s^2 in this case). Given that the mass of the block is 10.0 kg, the gravitational force would be F_gravity = 10.0 kg * 9.8 m/s^2 = 98 N.

2. Calculate the work done by the gravitational force.
- The work done by gravity can be calculated using the formula: W_gravity = F_gravity * h, where W_gravity is the work done by gravity, F_gravity is the gravitational force, and h is the height difference between points A and C. In this case, the height difference is not mentioned, so we assume it to be zero since the block comes to rest at point C.

3. Calculate the work done by the spring.
- The work done by the spring force can be calculated using the formula: W_spring = (1/2) * k * x^2, where W_spring is the work done by the spring, k is the force constant of the spring (2300 N/m), and x is the compression of the spring (0.200 m in this case).

4. Determine the net work done on the block.
- Since the block comes to rest at point C, the net work done on the block should be zero.

5. Determine the work done by friction between points B and C.
- The work done by friction can be calculated using the formula: W_friction = -F_friction * d, where W_friction is the work done by friction, F_friction is the frictional force, and d is the distance between points B and C (6.00 m in this case). The negative sign indicates that the work done by friction is in the opposite direction of displacement.

6. Set up the equation for the net work done.
- Since the net work done on the block is zero, the sum of the work done by gravity, the work done by the spring, and the work done by friction should be zero: W_gravity + W_spring + W_friction = 0.

7. Solve the equation for the coefficient of kinetic friction.
- Rearrange the equation to solve for the coefficient of kinetic friction: F_friction = (W_gravity + W_spring)/d. Plug in the known values for W_gravity, W_spring, and d, and then divide by the known mass of the block to get the coefficient of kinetic friction (μ_k).

By following these steps, you should be able to find the coefficient of kinetic friction between the block and the rough surface between points B and C.