You push a 4.6 kg block against a horizontal spring, compressing the spring by 27 cm. Then you release the block, and the spring sends it sliding across a tabletop. It stops 63 cm from where you released it. The spring constant is 250 N/m. What is the coefficient of kinetic friction between the block and the table?

Get the initial velocity Vo using conservation of energy.

(1/2)MVo^2 = (1/2)kX^2
(X = 0.27 m is the spring compression; k is the spring constant)

After you have Vo, use KE = work done to stop.

(1/2) M Vo^2 = Uk*M*g

Actually you can skip the Vo step and just use
(1/2)*k*X^2 = Uk*M*g

Solve for Uk, the kinetic friction coefficient

To find the coefficient of kinetic friction between the block and the table, we can start by analyzing the forces involved in the motion of the block.

First, let's consider the force exerted by the spring on the block when it is released. This force can be calculated using Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position:

F = -kx

where F is the force, k is the spring constant, and x is the displacement of the spring. In this case, the spring is compressed by 27 cm, so the displacement x is -0.27 m (taking into account that compression is negative in this case). Thus, we can calculate the force exerted by the spring as:

F = -kx = -(250 N/m)(-0.27 m) = 67.5 N

The force exerted by the spring is acting in the opposite direction to the motion of the block. Let's call this force Fs.

Next, let's consider the gravitational force acting on the block. The weight of the block can be calculated using the equation:

Fg = mg

where Fg is the gravitational force, m is the mass of the block, and g is the acceleration due to gravity. In this case, the mass of the block is 4.6 kg, and g is approximately 9.8 m/s^2. Thus, the gravitational force is:

Fg = (4.6 kg)(9.8 m/s^2) = 45.08 N

The gravitational force is acting vertically downwards. Let's call this force Fg.

Now, let's consider the frictional force acting on the block as it slides across the tabletop. The frictional force can be calculated using the equation:

Ff = μkN

where Ff is the frictional force, μk is the coefficient of kinetic friction, and N is the normal force. The normal force is equal in magnitude but opposite in direction to the gravitational force, so N = -Fg.

Now, let's consider the motion of the block. The net force acting on the block is the sum of the forces exerted by the spring and the frictional force:

ΣF = Fs + Ff + Fg

Since the block is not accelerating during the motion (it eventually comes to a stop), the net force is equal to zero. Thus, we have:

0 = Fs + Ff + Fg

Substituting the known values, we have:

0 = 67.5 N + μk(-Fg) + 45.08 N

0 = 67.5 N - μk(45.08 N) + 45.08 N

Simplifying this equation, we find:

-67.5 N = - μk(45.08 N)

To solve for the coefficient of kinetic friction (μk), we rearrange the equation:

μk = -67.5 N / -45.08 N

μk ≈ 1.4994

The coefficient of kinetic friction between the block and the table is approximately 1.4994.