When the work done is 100% (maximum work), the angle between the applied force and the displacement is 0 degrees (θ = 0°) or, in other words, the force and displacement are perfectly aligned.
To find the angle of the applied force with the displacement when the work done is 50% (half of the maximum work), we can use the concept of cosine of the angle between two vectors. The work done (W) is given by:
W = F * d * cos(θ)
where:
- W is the work done,
- F is the magnitude of the applied force,
- d is the magnitude of the displacement, and
- θ is the angle between the applied force and the displacement.
Since we want the work done to be 50% (half), we can express it as a fraction of the maximum work (W_max):
W = 0.5 * W_max
Since the cosine of 0° is 1, we can rewrite the equation as:
0.5 * W_max = F * d * cos(θ)
Since we are interested in finding the angle (θ), we can rearrange the equation as follows:
cos(θ) = (0.5 * W_max) / (F * d)
Now we can solve for θ by taking the inverse cosine (arccos) of both sides of the equation:
θ = arccos((0.5 * W_max) / (F * d))
Please note that the value of θ will depend on the magnitudes of the applied force (F) and the displacement (d), as well as the maximum work (W_max) for the given system or situation.