Question

A car pulls a trailer down the highway. Let \(F_{\mathrm{t}}\) be the magnitude of the force on the trailer due to the car, and let \(F_{c}\) be the magnitude of the force on the car due to the trailer. If the car and trailer are moving at a constant velocity across level ground, then \(F_{t}=F_{c}\). If the car and trailer are accelerating up a hill, what is the relationship between the two forces?

Short answer

#Answer# In general, the magnitudes of the forces acting on the car and the trailer are not equal when they are accelerating up a hill. The relationship between the forces depends on the masses of the car and trailer, the angle of the hill, and their acceleration.

Step-by-step solution

Analyze the forces on the car and trailer

There are two main forces acting on the car and trailer in this situation: (1) the gravitational force that pulls both the car and the trailer down the hill, and (2) the force exerted by the car to pull the trailer up the hill. Let's assume that the angle of the hill from the horizontal is \(\theta\). The gravitational force acting on the car and the trailer can be written as \(F_{g_c} = m_c g \sin(\theta)\) and \(F_{g_t} = m_t g \sin(\theta)\), where \(m_c\) and \(m_t\) are the masses of the car and trailer, and \(g\) is the acceleration due to gravity.

Apply Newton's second law to the car and the trailer

To find the net force on the car (\(F_{net_c}\)) and trailer (\(F_{net_t}\)), we'll apply Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (\(F = m \cdot a\)). For the car: \(F_{net_c} = m_c a_c = F_c - F_{g_c}\) For the trailer: \(F_{net_t} = m_t a_t = F_t - F_{g_t}\)

Determine the acceleration of the car and the trailer

Since the car and the trailer are connected, their accelerations will be the same, i.e., \(a_c = a_t = a\). Therefore, we can rewrite the two equations above as follows: \(m_c a = F_c - m_c g \sin(\theta)\) \(m_t a = F_t - m_t g \sin(\theta)\)

Determine the relationship between \(F_c\) and \(F_t\)

Now let's solve for the forces \(F_c\) and \(F_t\): \(F_c = m_c a + m_c g \sin(\theta)\) \(F_t = m_t a + m_t g \sin(\theta)\) To compare the magnitudes of these forces, we'll divide the equation for \(F_c\) by the equation for \(F_t\): \(\frac{F_c}{F_t} = \frac{m_c a + m_c g \sin(\theta)}{m_t a + m_t g \sin(\theta)}\) From this equation, we can see that the relationship between the forces \(F_c\) and \(F_t\) depends on the masses of the car and trailer, as well as the angle of the hill and their acceleration. In general, \(F_c\) and \(F_t\) are not equal when the car and trailer are accelerating up a hill.

Key concepts

Force and Acceleration

Understanding the relationship between force and acceleration is crucial when studying motion. According to Newton's second law, the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass. The formula expressing this law is
\[\begin{equation} F = ma \rend{equation} where \( F \) is the net force applied to the object, \( m \) is the mass of the object, and \( a \) is the acceleration. This law implies that for a constant mass, increasing the net force will result in a higher acceleration, and vice versa.
When a car pulls a trailer, the force exerted by the car onto the trailer (\( F_{\mathrm{t}} \)) dictates the acceleration of both the car and the trailer, assuming they are connected and thus move as one system. If this force increases, the acceleration increases, assuming mass remains constant. Conversely, at a constant velocity, the net force is zero, since acceleration is not present. This is why when the car and trailer move across level ground at a constant velocity, \( F_{\mathrm{t}} = F_{c} \), where \( F_{c} \) is the force on the car due to the trailer.

Gravitational Force

Gravitational force, also known as gravity, is a force of attraction that acts between two masses. On the surface of the Earth, this force causes objects to be attracted towards the center of the planet, and is what gives weight to physical objects. Newton's law of universal gravitation defines the gravitational force (\( F_{g} \)) between two masses as\[\begin{equation} F_{g} = G\frac{m_1m_2}{d^2} \rend{equation} where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of two objects, and \( d \) is the distance between the centers of their masses.
For objects near the Earth's surface, this simplifies to \( F_{g} = mg \), where \( m \) is the mass of the object, and \( g \) is the acceleration due to Earth's gravity (approximately \( 9.81 \) m/s^2). When a car and trailer ascend a hill, both are subjected to gravitational forces pulling them downhill, which are factored into the overall force calculations. The component of gravitational force acting parallel to the inclined hill on each can be expressed as \( m g \sin(\theta) \), where \( \theta \) is the hill's angle from the horizontal. This force opposes the forward force and must be overcome by the pulling force of the car for the system to accelerate uphill.

Constant Velocity

An object is said to be moving at a constant velocity if it maintains the same speed and direction over time. According to Newton's first law of motion, also referred to as the law of inertia, an object at rest will stay at rest and an object in motion will continue to move at a constant velocity unless acted upon by a net external force.
When the car and trailer discussed in our example move across level ground at a constant velocity, it illustrates a state of dynamic equilibrium. This means the forces acting on the car and trailer are balanced, resulting in no net force (\( F_{net} = 0 \)) and therefore no acceleration (\( a = 0 \)). In this scenario, even if forces are present (i.e., the car pulling the trailer and the trailer exerting an equal and opposite force on the car), they are equal in magnitude and opposite in direction, effectively cancelling each other out.
Understanding this principle is crucial when discussing motion as it tells us that in the absence of oppositional forces like friction or air resistance, no additional force would be needed for the car and trailer to maintain their state of motion, confirming the intrinsic relationship between force, velocity, and acceleration.