A 20 kg block on a horizontal surface is attached to a horizontalspring of spring constant k  4.0 kN/m. The block is pulledto the right so that the spring is stretched 10 cm beyond its relaxedlength, and the block is then released from rest.The frictional forcebetween the sliding block and the surface has a magnitude of 80 N.(a) What is the kinetic energy of the block when it has moved 2.0cm from its point of release? (b) What is the kinetic energy of theblock when it first slides back through the point at which the springis relaxed? (c) What is the maximum kinetic energy attained by theblock as it slides from its point of release to the point at which thespring is relaxed?

To tackle this problem, we need to analyze the energy transformations that occur as the block moves. We will consider the potential energy stored in the spring, the work done against friction, and the kinetic energy of the block at various points in its motion. Let's break it down step by step.

Understanding the System

We have a block with a mass of 20 kg attached to a spring with a spring constant of 4.0 kN/m (or 4000 N/m). The spring is stretched 10 cm (0.1 m) from its relaxed position, and the block is released from rest. The frictional force acting on the block is 80 N.

Calculating Initial Potential Energy

When the spring is stretched, it stores potential energy given by the formula:

Potential Energy (PE) = 0.5 * k * x²

Where:

• k = spring constant (4000 N/m)
• x = displacement from the relaxed position (0.1 m)

Substituting the values:

PE = 0.5 * 4000 N/m * (0.1 m)² = 20 J

Part (a): Kinetic Energy After Moving 2.0 cm

When the block moves 2.0 cm (0.02 m) from its release point, the spring is now stretched by 8.0 cm (0.08 m). We need to calculate the new potential energy:

New PE = 0.5 * k * (0.08 m)²

Calculating this gives:

New PE = 0.5 * 4000 N/m * (0.08 m)² = 12.8 J

The work done against friction as the block moves 2.0 cm is:

Work = Frictional Force * Distance = 80 N * 0.02 m = 1.6 J

Now, we can find the kinetic energy (KE) at this point using the conservation of energy:

Total Energy = Initial PE - Work against friction

KE = Initial PE - New PE - Work

KE = 20 J - 12.8 J - 1.6 J = 5.6 J

Part (b): Kinetic Energy at the Relaxed Position

When the block slides back through the point where the spring is relaxed, the spring potential energy is 0. The total energy at this point is equal to the initial potential energy minus the work done against friction over the entire distance of 10 cm (0.1 m):

Work against friction for 10 cm = 80 N * 0.1 m = 8 J

Thus, the kinetic energy at the relaxed position is:

KE = Initial PE - Work against friction

KE = 20 J - 8 J = 12 J

Part (c): Maximum Kinetic Energy Attained

The maximum kinetic energy occurs just as the block passes through the relaxed position. This is the same value we calculated in part (b), which is:

Maximum KE = 12 J

Summary of Results

• Part (a): Kinetic Energy after moving 2.0 cm = 5.6 J
• Part (b): Kinetic Energy at the relaxed position = 12 J
• Part (c): Maximum Kinetic Energy = 12 J

By analyzing the energy changes and the work done against friction, we can see how the kinetic energy varies as the block moves. This approach not only helps in solving the problem but also reinforces the principles of energy conservation in mechanical systems.