A 1200 kg car is being driven up a 5:0 hill. The frictional force is directed opposite to

the motion of the car and has a magnitude of f = 524 N. A force ~F is applied to the
car by the road and propels the car forward. In addition to these two forces, two other
forces act on the car: its weight ~W and the normal force ~FN directed perpendicular
to the road surface. The length of the road up the hill is 290 m. What should be the
magnitude of ~F, so that the net work done by all the forces acting on the car is 150
kJ?

To find the magnitude of force F needed to achieve a net work of 150 kJ, we need to consider the work done by all the forces acting on the car.

The work done by a force can be calculated by multiplying the magnitude of the force by the displacement of the object in the direction of the force.

In this problem, there are several forces involved: the force of friction (f), the weight of the car (W), the normal force (FN), and the force applied by the road (F).

The work done by the force of friction is given by:

Work_friction = f * displacement

Since the frictional force is directed opposite to the motion of the car, the displacement is negative. So the work done by friction is:

Work_friction = -f * displacement

Similarly, the work done by the weight of the car and the normal force is given by:

Work_weight = W * displacement
Work_normal = FN * displacement

Since the weight and normal force are perpendicular to the motion, they do not contribute to the net work done. Hence, their work is zero.

The net work done by all the forces is the sum of the work done by each force:

NetWork = Work_friction + Work_weight + Work_normal + Work_applied

Given that the net work is 150 kJ, we can set up the equation:

150 kJ = -f * displacement + W * displacement

To solve this, we need to find the displacement of the car up the hill. The length of the road up the hill is given as 290 m.

Now, we can calculate the displacement:

displacement = 290 m

Substituting the values into the equation, we have:

150 kJ = -524 N * 290 m + (1200 kg * 9.8 m/s^2) * 290 m

Convert the kJ to J, and perform the calculation:

150,000 J = -524 N * 290 m + (1200 kg * 9.8 m/s^2) * 290 m

150,000 = -524 * 290 + 1200 * 9.8 * 290

Solve for the magnitude of force F by rearranging the equation:

F = (150,000 + 524 * 290) / (9.8 * 290)

Evaluate the expression to get the magnitude of force F.