Final answer:
The magnitude of force P must be set to balance the work done by kinetic friction, which considers the angle of P and the coefficient of kinetic friction. Using equilibrium of forces, we can find the horizontal component of P that matches the force of friction and solve for P.
Step-by-step explanation:
To determine the magnitude of force P that must be applied to make the net work done by it and the kinetic frictional force zero, we can use the concept of work and the work-energy theorem. The net work done is the sum of the work done by all forces acting on an object. If the net work is zero, the kinetic energy of the crate remains constant, implying that the crate is moving at a constant velocity. Considering the forces acting on the crate, we have the gravitational force (weight), the normal force, the applied force P, and the kinetic frictional force
The kinetic frictional force is given by = μk * N, where μk is the coefficient of kinetic friction, and N is the normal force. In this case, the normal force is affected by the vertical component of force P because the force is applied at an angle. The normal force N equals the weight of the crate minus the vertical component of P, i.e., N = mg - P * sin(θ), where m is the mass of the crate, g is the acceleration due to gravity, and θ is the angle of the applied force below the horizontal.
For work done by the kinetic frictional force to be zero, the work done by the applied force P should be equal and opposite to the work done by friction. Thus, the magnitude of force P should be set such that the horizontal component of P overcomes the frictional force. Mathematically, P * cos(θ) =, and substituting for we get P * cos(θ) = μk * (mg - P * sin(θ)). Solving this equation will give us the magnitude of P.