Lazydog and his 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? (N)
To find the magnitude of the force in the coupling between the tractor and trailer, we need to consider the forces acting on the system.
1. The weight force acting on the tractor and trailer:
The weight force of the tractor is given by the formula:
W_tractor = m_tractor * g
where m_tractor is the mass of the tractor and g is the acceleration due to gravity. Similarly, the weight force of the trailer is given by:
W_trailer = m_trailer * g
where m_trailer is the mass of the trailer.
2. The force of friction between the tractor and the trailer:
Since the trailer wheels roll freely, there is no direct frictional force acting on them. However, there is a force of friction acting on the tractor due to the incline of the hill. This force of friction can be calculated using the formula:
F_friction = m_total * a
where m_total is the combined mass of the tractor and trailer, and a is the acceleration due to the incline. In this case, the acceleration due to the incline can be calculated using:
a = g * sin(θ)
where θ is the angle of the hill (7° in this case).
3. The net force acting on the system:
The net force acting on the system is given by:
F_net = F_couple - F_friction
where F_couple is the force in the coupling between the tractor and the trailer.
4. Applying Newton's second law:
According to Newton's second law, the net force (F_net) acting on an object is equal to its mass (m) multiplied by its acceleration (a):
F_net = m * a
Since the velocity is given to be decreasing, the acceleration (a) is equal to the negative of the given deceleration rate (0.2 m/s^2).
Now, let's plug in the given values and calculate the force in the coupling between the tractor and the trailer:
Given:
m_tractor = 1330 kg
m_trailer = 530 kg
θ = 7°
v = 8.7 m/s
deceleration (a) = -0.2 m/s^2
1. Calculate the weight forces:
W_tractor = m_tractor * g
W_trailer = m_trailer * g
2. Calculate the acceleration due to the incline:
a = g * sin(θ)
3. Calculate the force of friction:
F_friction = m_total * a
4. Calculate the net force:
F_net = m_total * a (since the acceleration is in the opposite direction to the motion)
5. Rearrange the equation to solve for F_couple:
F_couple = F_net + F_friction
Plug in all the calculated and given values to find the magnitude of the force in the coupling between the tractor and trailer (F_couple).