You push a 2.5 kg block against a horizontal spring, compressing the spring by 16 cm. Then you release the block, and the spring sends it sliding across a tabletop. It stops 60 cm from where you released it. The spring constant is 200 N/m. What is the block-table coefficient of kinetic friction?

0.174

To find the block-table coefficient of kinetic friction, we need to use the concept of conservation of mechanical energy. Here are the steps to find the coefficient of kinetic friction:

Step 1: Determine the potential energy of the compressed spring.
The potential energy stored in a spring can be calculated using the formula:
Potential energy (PE) = 0.5 * k * x^2
Where k is the spring constant (200 N/m) and x is the compression of the spring (16 cm = 0.16 m).
Plugging in the values, we get:
PE = 0.5 * 200 * (0.16^2) = 3.2 J

Step 2: Find the initial kinetic energy of the block-table system.
When the block is released, the potential energy of the spring is converted into kinetic energy of the block. Therefore, the initial kinetic energy of the block-table system is equal to the potential energy of the spring (as there is no other external force in play).
Initial kinetic energy (KE) = 3.2 J

Step 3: Find the work done against friction.
The work done against friction can be calculated using the formula:
Work done against friction = Force of friction * Distance
The work done against friction is equal to the loss in kinetic energy.
In this case, the work done against friction is equal to the initial kinetic energy (3.2 J) minus the final kinetic energy, as the block comes to a stop (0 J).

Step 4: Find the force of friction.
The force of friction can be found using the formula:
Force of friction = Work done against friction / Distance
Plugging in the values, we get:
Force of friction = 3.2 J / 0.6 m = 5.33 N

Step 5: Find the normal force.
The normal force acting on the block is equal to its weight (mg) as there is no vertical acceleration.
The weight of the block can be found using the formula:
Weight = mass * acceleration due to gravity (g)
In this case, the mass of the block is 2.5 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
Weight = 2.5 kg * 9.8 m/s^2 = 24.5 N

Step 6: Calculate the coefficient of kinetic friction.
The coefficient of kinetic friction can be found using the formula:
Coefficient of kinetic friction = Force of friction / Normal force
Plugging in the values, we get:
Coefficient of kinetic friction = 5.33 N / 24.5 N = 0.217

Therefore, the block-table coefficient of kinetic friction is approximately 0.217.