Final answer:
To find the tension in the rope needed to pull an 8.0 kg crate up a 30° incline considering kinetic friction, decompose the gravitational force and the tension into components parallel and perpendicular to the incline plane, apply Newton's second law for motion along the incline, and solve for tension using the conditions of constant velocity (zero acceleration).
Step-by-step explanation:
To calculate the tension in the rope when pulling an 8.0 kg crate up a 30° incline, consider the forces acting on the crate. These include the gravitational force, normal force, frictional force, and the force from the rope. The gravitational force can be decomposed into components perpendicular and parallel to the incline. The frictional force is determined by the coefficient of kinetic friction and the normal force.
The tension in the rope can then be calculated using Newton's second law, where the sum of the forces equals the mass times the acceleration, but since the crate is pulled up at a constant velocity, the acceleration will be zero. Thus, the sum of the forces in the direction of the incline will also be zero. By setting up the equation and solving for the tension, you can find the correct value.
The rope's angle relative to the incline also needs to be accounted for; this changes the direction of the force exerted by the rope. Trigonometry helps decompose this force into parallel and perpendicular components to the incline, which then are used to find the net force and consequently the tension in the rope.