A 1310-kg car is being driven up a 9.12 ° hill. The frictional force is directed opposite to the motion of the car and has a magnitude of 541 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 337 m. What should be the magnitude of F, so that the net work done by all the forces acting on the car is 155 kJ?
Wc = m * g = 1310kg * 9.8N/kg = 12838 N.
Fp = 12338*sin9.12o = 2035 N. = Force
parallel to road.
Fv = 12838*cos9.12 = 12676 N. = Force
perpendicular to road. = The normal.
Work = Fn * D = 155,00 J.
Fn * 337 = 155000
Fn = 460 N. = Net force.
Fn = F-Fp-Fk = 460 N.
F-2035-541 = 460
F = 460+2035+541 = 3036 N.
To determine the magnitude of force F, we need to calculate the net work done by all the forces acting on the car.
The net work done is given by the equation: W_net = ΔKE = KE_final - KE_initial, where KE is the kinetic energy of the car.
Since the car is being driven up a hill, it initially has no kinetic energy (KE_initial = 0). The final kinetic energy (KE_final) can be calculated using the work-energy theorem:
W_net = ΔKE (1)
W_net = KE_final - KE_initial
W_net = KE_final - 0
W_net = KE_final
Let's start by calculating the work done by the frictional force. The work done by a force is given by the formula: W = F * d * cosθ, where F is the force, d is the displacement, and θ is the angle between the force and the displacement.
Given:
Frictional force magnitude (f) = 541 N
Length of the road up the hill (d) = 337 m
θ = 180° (since the frictional force is opposite to the motion of the car)
Work done by frictional force (W_friction) = f * d * cosθ
W_friction = 541 N * 337 m * cos(180°)
Now, let's calculate the gravitational potential energy of the car. The gravitational potential energy is given by the equation: PE = m * g * h, where m is the mass of the car, g is the acceleration due to gravity, and h is the vertical height.
Given:
Mass of the car (m) = 1310 kg
Angle of the hill (θ_hill) = 9.12°
Height (h) = d * sin(θ_hill) (using trigonometry to find the vertical height)
Gravitational potential energy (PE) = m * g * h
PE = 1310 kg * 9.8 m/s^2 * (337 m * sin(9.12°))
Now, we can substitute the values and calculate the net work done by all the forces:
W_net = KE_final = W_friction - PE
Since the net work is given as 155 kJ, we can convert it to Joules:
W_net = 155 kJ = 155,000 J
Now, substitute the calculated values and solve for F:
155,000 J = (W_friction) - (PE)
Therefore,
F = W_net + PE - W_friction
Substitute the values of W_net, PE, and W_friction and perform the calculation to find the magnitude of force F.