A crate is dragged 3 meters along a smooth level floor by a 30 N force, applied at 25° to the floor. Then, it is pulled 4 meters up a ramp inclined at 20° to the horizontal, using the same force. Then, the crate is dragged a further 5 meters along a level platform using the same force again. Determine the total work done in moving the crate.
To determine the total work done in moving the crate, we need to calculate the work done in each segment of the movement - dragging along the smooth floor, pulling up the ramp, and dragging along the level platform.
1. Work done while dragging along the smooth floor:
The formula for work is given by W = Fd cosθ, where W represents work done, F is the force applied, d is the displacement, and θ is the angle between the force and displacement vectors. In this case, the force applied is 30 N and the displacement is 3 meters. The angle between the force and displacement vectors can be found using trigonometry as cosθ = cos(180° - θ). Since the force is applied at 25° to the floor, the angle between the force and displacement vectors is 180° - 25° = 155°. Plugging the values into the formula, we have:
W1 = 30 N * 3 m * cos(155°)
2. Work done while pulling up the ramp:
Similar to the first step, we use the formula W = Fd cosθ. The force and displacement remain the same at 30 N and 4 meters, respectively. The angle between the force and displacement vectors is 90° - 20° = 70°. Applying the formula, we get:
W2 = 30 N * 4 m * cos(70°)
3. Work done while dragging along the level platform:
Again, using the formula W = Fd cosθ, the force and displacement remain the same (30 N and 5 meters). The angle between the force and displacement vectors is 0° since they are parallel. Thus, the angle of cosθ is 0°, and we have:
W3 = 30 N * 5 m * cos(0°)
Now, to find the total work done, we simply sum up the work done in each segment:
Total work done = W1 + W2 + W3
Note that the cosine function can be evaluated using a scientific calculator or trigonometric tables.