A 0.565 kg wood block is firmly attached to a very light horizontal spring (k = 185 N/m) as shown in Fig. 6-40. It is noted that the block-spring system, when compressed 5.0 cm and released, stretches out 2.3 cm beyond the equilibrium position before stopping and turning back. What is the coefficient of kinetic friction between the block and the table?
To find the coefficient of kinetic friction between the block and the table, we can use the concept of energy conservation.
In this case, when the block-spring system is compressed and then released, the potential energy is converted into kinetic energy as the block moves. When the block comes to a stop and turns back, all the energy has been dissipated due to the work done against friction.
Let's break down the problem into steps:
Step 1: Calculate the potential energy stored in the spring when it is compressed.
The potential energy stored in a spring is given by the formula:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the compression or extension of the spring.
In this case, the block is compressed by 5.0 cm. So, the potential energy stored in the spring is:
Potential Energy = (1/2) * 185 N/m * (5.0 cm)^2
Step 2: Determine the work done against friction.
The work done against friction can be calculated using the formula:
Work = force of friction * distance
where the force of friction can be determined using the equation:
force of friction = coefficient of friction * normal force
and the normal force is equal to the weight of the block.
The weight of the block can be calculated as:
Weight = mass * gravity
where the mass is 0.565 kg and gravity is 9.8 m/s^2.
So, the normal force is:
Normal force = 0.565 kg * 9.8 m/s^2
Step 3: Relate the work done against friction to the energy dissipated.
When the block comes to a stop and turns back, all the energy has been dissipated due to the work done against friction. This means the work done against friction is equal to the initial potential energy of the spring.
Therefore, we can equate the work done against friction to the potential energy stored in the spring:
Work = (1/2) * 185 N/m * (5.0 cm)^2
Step 4: Solve for the coefficient of friction.
Now, we can solve for the coefficient of friction. Rearranging the equation from step 3, we have:
coefficient of friction = Work / (Normal force * distance)
Substituting the values we've calculated, we can find the coefficient of friction.