A 6 kg block free to move on a horizontal, frictionless surface is attached to a spring. The spring is compressed 0.16 m from equi- librium and released. The speed of the block is 1.44 m/s when it passes the equilibrium po- sition of the spring. The same experiment is now repeated with the frictionless surface re- placed by a surface with coefficient of friction 0.1.

Determine the speed of the block at the equilibrium position of the spring. The accel- eration due to gravity is 9.8 m/s2 .

To determine the speed of the block at the equilibrium position of the spring in this scenario, we need to consider energy conservation.

First, let's consider the scenario with the frictionless surface.

The block is attached to a spring, so we can assume it follows simple harmonic motion. At the equilibrium position of the spring, the potential energy stored in the spring is zero because it is neither compressed nor stretched.

The initial mechanical energy of the system when the block is compressed can be calculated as follows:

Initial Mechanical Energy = Potential Energy + Kinetic Energy
= 0 + 0.5 * mass * velocity^2

The mass of the block is 6 kg, and the velocity at this point is 1.44 m/s.

Now, let's consider the scenario with the frictional surface.

The presence of friction will cause a loss of mechanical energy as the block moves from its initial position to the equilibrium position of the spring.

The work done against friction can be calculated using the formula:

Work = Force of Friction * Distance

The force of friction can be calculated using the formula:

Force of Friction = Coefficient of Friction * Normal Force

The normal force in this case is equal to the weight of the block, which can be calculated using the formula:

Weight = mass * acceleration due to gravity

The coefficient of friction is given as 0.1, and the distance over which the block moves is the same as the compression of the spring, which is 0.16 m.

So, the work done against friction is:

Work = (Coefficient of Friction * mass * acceleration due to gravity) * Distance

Now, we can use energy conservation again.

Final Mechanical Energy = Initial Mechanical Energy - Work

At the equilibrium position of the spring, all of the mechanical energy is in the form of kinetic energy because the potential energy of the spring is zero.

Finally, we can calculate the speed of the block at the equilibrium position by rearranging the kinetic energy formula:

Final Mechanical Energy = 0.5 * mass * velocity^2

Given all the values, you can substitute them into the equations and solve for the final velocity at the equilibrium position of the spring in both scenarios.