Final answer:
The magnitude of the force vector needed to maintain a constant velocity for a crate on an incline is calculated by balancing the parallel component of gravitational force and the force of friction, which are opposing the applied horizontal force.
Step-by-step explanation:
To find the magnitude of the force vector needed to keep a crate moving at constant velocity on an incline, one must consider the forces acting on the crate. These include the force of gravity, the normal force, the force of friction, and the applied force. Since the crate is moving at constant velocity, the net force must be zero according to Newton's first law of motion. Therefore, the applied force must be equal in magnitude and opposite in direction to the sum of the other forces.
Starting with the force of gravity, we can calculate the component of the weight of the crate parallel to the incline (mg sin(θ)). Then, we calculate the force of friction, which is the product of the normal force (mg cos(θ)) and the coefficient of kinetic friction (μk). The magnitude of the applied force F required to maintain constant velocity will be equal to the sum of these two forces, F = mg sin(θ) + μkmg cos(θ).
For example, for a crate with a mass of 40kg on a 30° incline, with a coefficient of kinetic friction of 0.50, the force required would be:
F = (40kg)(9.8m/s²) sin(30°) + (0.50)(40kg)(9.8m/s²) cos(30°). Simplifying, the magnitude of F can be solved to maintain constant velocity.