In the figure below, a constant force of magnitude 82.0 N is applied to a 2.55 kg shoe box at angle = 64.5°, causing the box to move up a frictionless ramp at constant speed. How much work is done on the box by a when the box has moved through vertical distance h = 0.180 m?

Since the applied force causes the box to move at a constant speed implies that there are no changes in KE => The work done by Fa = change in PE:

W(a) = mgh= 2.55•9.8•0.18 = 4.5 J.

To find the work done on the box by the force, we need to determine the component of the force in the direction of motion and then multiply it by the distance moved.

Step 1: Find the component of the force in the direction of motion.

The force can be split into two components: one perpendicular to the ramp and one parallel to the ramp. The component parallel to the ramp is responsible for the motion of the box.

F_parallel = F * cos(theta)
F_parallel = 82.0 N * cos(64.5°)

Step 2: Calculate the work done.

The work done can be determined using the formula:

Work = Force * Distance

Work = F_parallel * h
Work = (82.0 N * cos(64.5°)) * 0.180 m

Now, we can calculate the value.

Work = (82.0 N * cos(64.5°)) * 0.180 m
Work ≈ 735.6 J

Therefore, the work done on the box is approximately 735.6 Joules.

To find the work done on the shoe box, we need to use the formula:

Work = Force * Distance * cos(theta)

Where:
- Work is the amount of work done
- Force is the magnitude of the applied force
- Distance is the distance over which the force is applied
- theta is the angle between the force and the displacement

First, let's find the distance the box moves along the ramp. We can do this by using the given vertical displacement, h, and the angle of the ramp.

Distance along the ramp = h / sin(theta)

Plugging in the given values:
Distance along the ramp = 0.180 m / sin(64.5°)

Next, we can calculate the work done by substituting the values into the formula:

Work = (82.0 N) * (distance along the ramp) * cos(64.5°)

Calculating the distance along the ramp and the cosine of the angle:

Distance along the ramp = 0.180 m / sin(64.5°) = 0.204 m
cos(64.5°) = 0.438

Now we can substitute these values back into the formula:

Work = (82.0 N) * (0.204 m) * (0.438)

Calculating the work:

Work ≈ 7.006 J

Therefore, the work done on the box by the applied force is approximately 7.006 Joules.