An animal and its tractor (1330 kg) are pulling a trailer (530 kg) up a 7° hill. The trailer wheels roll freely. What is the magnitude of the force in the coupling between the tractor and trailer when their velocity is 8.7 m/s up the hill and they are slowing down at a rate of 0.2 m/s2?
Is the animal driving the tractor?
yes
To solve this problem, we need to consider the forces acting on the tractor-trailer system and apply Newton's second law of motion.
First, let's calculate the gravitational force acting on the tractor and trailer. The gravitational force can be calculated using the formula:
F_gravity = mass * gravity
Given that the mass of the tractor is 1330 kg and the mass of the trailer is 530 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the gravitational force acting on the system:
F_gravity = (1330 kg + 530 kg) * 9.8 m/s^2 = 18640 N
Next, let's consider the force slowing down the tractor-trailer system. This force can be calculated using Newton's second law of motion:
F_slowing = mass * acceleration
Given that the total mass of the system is 1330 kg + 530 kg = 1860 kg, and the acceleration is -0.2 m/s^2 (negative because the system is slowing down), we can calculate the force slowing down the system:
F_slowing = 1860 kg * (-0.2 m/s^2) = -372 N
The negative sign indicates that the force is acting in the opposite direction of motion.
Finally, to find the force in the coupling between the tractor and trailer, we need to consider the net force acting on the system:
Net force = F_traction + F_gravity + F_slowing
Since the system is moving up the hill, the traction force in the direction of motion is the force in the coupling between the tractor and trailer. The net force is given by the following equation:
Net force = mass * acceleration
Since the system is moving at a constant velocity of 8.7 m/s, the acceleration is 0, and therefore, the net force is 0. We can set up an equation to solve for the force in the coupling:
0 = F_traction + F_gravity + F_slowing
Rearranging the equation, we can solve for the force in the coupling:
F_traction = -(F_gravity + F_slowing)
Plugging in the values we calculated earlier:
F_traction = - (18640 N + (-372 N)) = -19012 N
The negative sign indicates that the force in the coupling between the tractor and trailer is acting in the opposite direction of motion, which makes sense since the system is slowing down.
Therefore, the magnitude of the force in the coupling between the tractor and trailer is 19012 N.