b) To determine the work done by the crane on the beam, we need to use the formula:
work = force × distance × cos(theta)
where force is the force exerted by the crane, distance is the distance the beam is raised, and theta is the angle between the force and the displacement.
Since the beam is raised at a constant velocity, the net force on the beam is zero, which means that the force exerted by the crane is equal in magnitude and opposite in direction to the force of gravity.
The force of gravity on the beam is given by:
force_gravity = mass × gravity
where mass is the mass of the beam and gravity is the acceleration due to gravity.
Substituting the given values, we get:
force_gravity = 250 kg × 9.81 m/s^2 = 2452.5 N
Since the net force on the beam is zero, the force exerted by the crane is also 2452.5 N, but in the opposite direction. Therefore, the work done by the crane on the beam is:
work = force × distance × cos(theta) = -2452.5 N × 25 m × cos(180°) = -61,312.5 J
The negative sign indicates that the work done by the crane is negative, which means that the crane is doing work against the force of gravity.
c) The work done by gravity on the beam is given by:
work_gravity = force_gravity × distance × cos(theta)
where force_gravity is the force of gravity on the beam, distance is the distance the beam is raised, and theta is the angle between the force and the displacement.
Substituting the given values, we get:
work_gravity = 2452.5 N × 25 m × cos(180°) = -61,312.5 J
The negative sign indicates that the work done by gravity is negative, which means that gravity is doing work against the motion of the beam. Note that the magnitude of the work done by gravity is the same as the magnitude of the work done by the crane, but with opposite sign, as expected due to the work-energy principle.