A 1,200-kg car is pulling a 305-kg trailer. Together, the car and trailer have an acceleration of 2.00 m/s2 in the positive x-direction. Neglecting frictional forces on the trailer, determine the following. (Indicate the direction with the sign of your answer. Take the forward direction to be positive. Assume the trailer's weight is entirely supported by it's own tires.)
(a) the net force on the car
N
(b) the net force on the trailer
N
(c) the force exerted by the trailer on the car
N
(d) the resultant force exerted by the car on the road
magnitude N
direction ° (below the horizontal from the rearward direction)
To solve this problem, we'll use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. We'll begin by calculating the net force on the car.
(a) The net force on the car:
To find the net force on the car, we'll use the formula:
F_net = m_car * a
where F_net is the net force, m_car is the mass of the car, and a is the acceleration.
Given:
m_car = 1,200 kg
a = 2.00 m/s^2
Plugging in the values, we get:
F_net = (1,200 kg) * (2.00 m/s^2)
= 2,400 kg·m/s^2
Therefore, the net force on the car is 2,400 N in the positive x-direction.
(b) The net force on the trailer:
Since the trailer is being pulled by the car, the net force on the trailer will be the same as the net force on the car. Therefore, the net force on the trailer is also 2,400 N in the positive x-direction.
(c) The force exerted by the trailer on the car:
To find the force exerted by the trailer on the car, we need to consider that the force exerted by the trailer must be equal to the net force on the car. Therefore, the force exerted by the trailer on the car is also 2,400 N in the positive x-direction.
(d) The resultant force exerted by the car on the road:
The resultant force exerted by the car on the road is the sum of the forces exerted by the car and the trailer. Since both forces are in the same direction, we can simply add their magnitudes:
Resultant force = |Force exerted by car| + |Force exerted by trailer|
Resultant force = 2,400 N + 2,400 N
= 4,800 N
The direction of the resultant force is given as "below the horizontal from the rearward direction." This means that the force is at an angle below the horizontal line. However, we don't have enough information to determine the exact angle without additional data.
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Given:
Mass of car (m1) = 1,200 kg
Mass of trailer (m2) = 305 kg
Acceleration (a) = 2.00 m/s^2
(a) To find the net force on the car, we can use the formula:
Net force on car = (mass of car) × (acceleration)
Net force on car = (1,200 kg) × (2.00 m/s^2) = 2,400 N
Therefore, the net force on the car is 2,400 N in the positive x-direction.
(b) Since the trailer is not accelerating in this problem, the net force on the trailer is zero. This is because the forces acting on the trailer are the gravitational force downwards and the force exerted by the car. These two forces cancel each other out.
Therefore, the net force on the trailer is 0 N.
(c) The force exerted by the trailer on the car is equal in magnitude but opposite in direction to the force exerted by the car on the trailer. This is due to Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.
Therefore, the force exerted by the trailer on the car is 2,400 N in the negative x-direction.
(d) To find the resultant force exerted by the car on the road (equal to the net force on the car), we need to consider the forces acting on the car.
The forces acting on the car are:
1. The force exerted by the trailer on the car (2,400 N in the negative x-direction)
2. The gravitational force acting on the car (the weight of the car)
Since the weight of the car is not given, we'll assume it is 9.8 m/s^2 (the acceleration due to gravity) multiplied by the mass of the car:
Weight of car = (mass of car) × (acceleration due to gravity)
Weight of car = (1,200 kg) × (9.8 m/s^2) = 11,760 N
The resultant force exerted by the car on the road can be found by subtracting the force exerted by the trailer from the weight of the car:
Resultant force exerted by the car = Weight of car - Force exerted by the trailer
Resultant force exerted by the car = 11,760 N - 2,400 N = 9,360 N
Therefore, the resultant force exerted by the car on the road has a magnitude of 9,360 N.
To find the direction of the resultant force, we need to calculate the angle between the resultant force and the horizontal. Since the force exerted by the trailer is in the negative x-direction and the weight of the car is in the negative y-direction, the resultant force will have a downward component and a rearward component.
The angle can be found using trigonometry. Let's call the angle theta.
tan(theta) = (rearward component of resultant force) / (downward component of resultant force)
tan(theta) = (Force exerted by the trailer) / (Weight of car)
tan(theta) = 2,400 N / 11,760 N
theta = arctan(2,400 N / 11,760 N) ≈ 11.8°
Therefore, the resultant force exerted by the car on the road has a magnitude of 9,360 N and is directed approximately 11.8° below the horizontal in the rearward direction.