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 65 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 2.8 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

(a) The position of the block is 0.042769 cans of silly string.

(b) The work that has been done on the block by the applied force is equivalent to the energy expended by a squirrel trying to crack a coconut with a toothpick. In other words, it's a lot of wasted effort.

(c) The work done on the block by the spring force is like trying to fill a swimming pool with a leaky bucket. In other words, not much.

(d) The block's position when its kinetic energy is maximum is at the point where it realizes it left the stove on at home and starts running back there.

(e) The value of that maximum kinetic energy is approximately 1 Whisker Biscuit (which is a unit of measurement only known to cats).

(a) To find the position of the block, we need to consider the force applied by the spring and the applied force. We know that the spring force acts in the opposite direction of the displacement, while the applied force acts in the same direction. At the stopping point, the net force on the block is zero.

The force due to the spring is given by Hooke's Law: F_spring = -kΔx, where k is the spring constant and Δx is the displacement from the relaxed length. The applied force is constant and has a magnitude of 2.8 N.

At the stopping point, the net force is zero, so we have:

F_spring + F_applied = 0
-65Δx + 2.8 = 0

Solving for Δx, we find:

Δx = 2.8 / 65 ≈ 0.043 m

Therefore, the position of the block is approximately 0.043 m.

(b) The work done on the block by the applied force can be found using the formula:

Work = Force × Displacement × cosθ

Since the applied force is in the same direction as the displacement, the angle θ between them is 0 degrees. Therefore, the cosine of 0 degrees is 1, and the work done is given by:

Work_applied = F_applied × Δx × cos0°
= 2.8 N × 0.043 m × 1
≈ 0.1204 J

Thus, the work done on the block by the applied force is approximately 0.1204 J.

(c) The work done on the block by the spring force can be found using the same formula as above, but substituting the force due to the spring for the applied force.

Work_spring = F_spring × Δx × cosθ

Since the spring force is in the opposite direction of the displacement, the angle θ between them is 180 degrees. Therefore, the cosine of 180 degrees is -1, and the work done is given by:

Work_spring = F_spring × Δx × cos180°
= -65 N/m × 0.043 m × (-1)
≈ 0.1395 J

Thus, the work done on the block by the spring force is approximately 0.1395 J.

(d) The block's position when its kinetic energy is maximum can be found using the work-energy theorem. The work done on an object is equal to the change in its kinetic energy. At the stopping point, the block's kinetic energy is zero, so the work done on the block is equal to its initial kinetic energy.

We know that work done by the applied force is 0.1204 J and the work done by the spring force is 0.1395 J. Since the applied force is responsible for stopping the block, the work done by the spring force is negative.

Therefore, the maximum kinetic energy of the block occurs when the work done by the applied force is equal to the magnitude of the work done by the spring force. This occurs when:

Work_applied = |Work_spring|

Substituting the previously calculated values, we have:

0.1204 J = |0.1395 J|

Thus, the block's position when its kinetic energy is maximum is at approximately 0.043 m.

(e) The maximum kinetic energy is equal to the work done by the applied force when the work done by the applied force is equal to the magnitude of the work done by the spring force.

Therefore, the maximum kinetic energy is equal to the magnitude of the work done by the spring force, which is:

|Work_spring| ≈ 0.1395 J

Thus, the value of the maximum kinetic energy is approximately 0.1395 J.

To analyze the given situation, we'll use the principles of work and energy.

(a) To find the position of the block when it stops, we need to use Hooke's Law. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

In this case, the spring constant is given as 65 N/m, and the applied force is 2.8 N. As the block is moving in the positive direction of the x-axis, the spring force will oppose the applied force.

Therefore, the spring force is -2.8 N (negative because it's in the opposite direction). We can use Hooke's Law to find the displacement of the block:

F = -kx, where F is the force, k is the spring constant, and x is the displacement.

-2.8 N = -65 N/m * x

Solving for x, we find:

x = 0.043 m

So, the position of the block when it stops is 0.043 m.

(b) To calculate the work done on the block by the applied force, we can use the formula:

Work = Force * Distance * cos(theta)

In this case, the force is 2.8 N and the distance is the displacement of the block, which is 0.043 m.

Since the applied force is in the positive x-direction, and the displacement is also in the positive x-direction, the angle (theta) between the force and displacement vectors is 0 degrees. So, cos(0) = 1.

Therefore, the work done on the block by the applied force is:

Work = 2.8 N * 0.043 m * cos(0) = 0.1204 J (Joules)

(c) The work done on the block by the spring force is equal in magnitude but opposite in direction to the work done by the applied force. So, the work done on the block by the spring force is -0.1204 J.

(d) The block's position when its kinetic energy is maximum can be found using the conservation of mechanical energy. The total mechanical energy of the system remains constant as there is no non-conservative force acting.

At the stopping point, the block comes to rest, so its total mechanical energy is entirely in the form of potential energy stored in the spring.

The potential energy stored in the spring is given by the formula:

Potential Energy = (1/2) * k * x^2

Substituting the values, we get:

Potential Energy = (1/2) * 65 N/m * (0.043 m)^2 = 0.0293 J

Now, to find the block's position when its kinetic energy is maximum, we need to find the point where the potential energy is minimized. This occurs when the displacement is 0.

So, when x = 0, the block's kinetic energy is maximum.

(e) Since the block is at rest when x = 0, its kinetic energy is zero at that point. Therefore, the maximum kinetic energy is 0 J.

F = k x so x = F/k

Work done by force = U stored in spring = (1/2) k x^2

Work done by spring = -(1/2) k x^2

you let thew block go for d I assume
then the minimum U is when x = 0 and that is the maximum speed and kinetic Energy

Max KE = U at max stretch x = (1/2) k x^2