The block in the figure below lies on a horizontal frictionless surface and is attached to the free end of the spring, with a spring constant of 55 N/m. Initially, the spring is at its relaxed length and the block is stationary at position x = 0. Then an applied force with a constant magnitude of 2.7 N pulls the block in the positive direction of the x axis, stretching the spring until the block stops.

Assume that the stopping point is reached.

(a) What is the position of the block?


m

(b) What is the work that has been done on the block by the applied force?
J

(c) What is the work that has been done on the block by the spring force?
J
During the block's displacement, find the following values.
(d) The block's position when its kinetic energy is maximum.
m

(e) The value of that maximum kinetic energy.
mJ

To answer these questions, we need to use the principles of work and energy.

(a) To find the position of the block, we know that the applied force does work on the block until it stops. Since no external forces are acting on the block horizontally, the work done by the applied force is equal to the change in potential energy of the spring. We can use Hooke's Law to relate the force and displacement:

F = -kx

Where F is the force, k is the spring constant, and x is the displacement from the equilibrium position. Solving for x, we have:

x = F / k
= 2.7 N / 55 N/m
= 0.049 m

So, the position of the block is 0.049 m.

(b) The work done by the applied force is given by the equation:

Work = force * displacement * cos(theta)

Since the applied force and displacement are in the same direction, the angle theta is 0 degrees and cos(theta) is 1. Plugging in the values, we get:

Work = 2.7 N * 0.049 m * cos(0)
= 0.1323 J

So, the work done by the applied force is 0.1323 J.

(c) The work done by the spring force can be calculated using the equation:

Work = (1/2) * k * x^2

Plugging in the values, we get:

Work = (1/2) * 55 N/m * (0.049 m)^2
= 0.066 J

So, the work done by the spring force is 0.066 J.

(d) The block's position when its kinetic energy is maximum can be found using the principle of conservation of mechanical energy. At the point where the block stops, all the work done by the applied force gets converted to potential energy. At this point, the block is momentarily at rest, so there is no kinetic energy. Hence, the maximum kinetic energy occurs at the initial position when the block is released.

Therefore, the block's position when its kinetic energy is maximum is at x = 0 m.

(e) Since the maximum kinetic energy occurs at the initial position, where the block is released and has no potential energy, we can calculate the maximum kinetic energy using the equation:

Kinetic energy = Work done by applied force

Plugging in the values, we get:

Kinetic energy = 0.1323 J

So, the value of the maximum kinetic energy is 0.1323 J.