A 1100-kg car is being driven up a 4.0° hill. The frictional force is directed opposite to the motion of the car and has a magnitude of f = 518 N. A force 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 and the normal force N directed perpendicular to the road surface. The length of the road up the hill is 310 m. What should be the magnitude of , so that the net work done by all the forces acting on the car is +140 kJ?
To solve this problem, we need to find the work done by each force acting on the car and then determine the value of the applied force F.
The work done by a force is given by the equation:
Work = Force * Distance * cos(θ)
Where:
- Work is the work done by the force in joules (J)
- Force is the magnitude of the force in newtons (N)
- Distance is the distance over which the force is applied in meters (m)
- θ is the angle between the force and the direction of motion in degrees
Let's calculate the work done by each force:
1. Frictional Force (opposite to motion):
- Force = Magnitude of frictional force = 518 N
- Distance = Length of road up the hill = 310 m
- θ = 180 degrees (opposite direction)
- Work_friction = 518 N * 310 m * cos(180°)
2. Weight Force (downward):
- Force = Weight = mass * acceleration due to gravity
- mass = 1100 kg
- acceleration due to gravity = 9.8 m/s^2
- Distance = 310 m
- θ = 90 degrees (perpendicular direction)
- Work_weight = (1100 kg * 9.8 m/s^2) * 310 m * cos(90°)
3. Applied Force (forward):
- Force = F (unknown)
- Distance = 310 m
- θ = 0 degrees (same direction)
- Work_applied = F * 310 m * cos(0°)
Total Work Done = Work_friction + Work_weight + Work_applied
We know that the net work done by all the forces is +140 kJ, which is equivalent to 140,000 J.
Therefore, we can write the equation:
140,000 J = Work_friction + Work_weight + Work_applied
Substituting the calculated values into the equation, we get:
140,000 J = (518 N * 310 m * cos(180°)) + ((1100 kg * 9.8 m/s^2) * 310 m * cos(90°)) + (F * 310 m * cos(0°))
Now, let's solve for F by rearranging the equation:
F * 310 m * cos(0°) = 140,000 J - (518 N * 310 m * cos(180°)) - ((1100 kg * 9.8 m/s^2) * 310 m * cos(90°))
Simplifying the equation:
F = (140,000 J - (518 N * 310 m * cos(180°)) - ((1100 kg * 9.8 m/s^2) * 310 m * cos(90°))) / (310 m * cos(0°))
Now, let's calculate the value of F.
To find the magnitude of the force that needs to be applied to the car, we can use the work-energy principle. According to the work-energy principle, the net work done on an object is equal to the change in its kinetic energy. In this case, the net work done on the car is equal to the sum of the work done by the applied force and the work done by the frictional force:
Net Work = Work done by applied force + Work done by frictional force
Given that the net work done is +140 kJ, we can convert this to joules:
Net Work = +140 kJ = +140,000 J
Now let's calculate the work done by the applied force. The work done by a force is equal to the dot product of the force and the displacement. The dot product of two vectors is given by the product of their magnitudes and the cosine of the angle between them:
Work done by applied force = |force| * |displacement| * cos(angle)
The magnitude of the displacement, |displacement|, is given as 310 m.
The angle between the applied force and the displacement is the same as the angle of the hill, which is 4.0°.
Now we can substitute the values into the equation to calculate the work done by the applied force:
Work done by applied force = |force| * |displacement| * cos(angle)
Work done by applied force = |force| * 310 m * cos(4.0°)
Given that the frictional force has a magnitude of 518 N, and it acts opposite to the motion of the car, we know that the work done by the frictional force is equal to the product of the magnitude of the frictional force and the displacement:
Work done by frictional force = -|frictional force| * |displacement|
Work done by frictional force = -518 N * 310 m
Now we can rewrite the equation for the net work done:
+140,000 J = |force| * 310 m * cos(4.0°) - 518 N * 310 m
To find the magnitude of the force needed to achieve the desired net work done, we can rearrange the equation:
|force| = (+140,000 J + 518 N * 310 m) / (310 m * cos(4.0°))
Now we can substitute the given values into the equation and calculate the magnitude of the force.