Question

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

Short answer

Answer: When the car and trailer are moving at a constant velocity on level ground, the forces are balanced, and F_t = F_c. However, when the car is accelerating up a hill, the relationship between F_t and F_c is affected by the masses of the car and trailer as well as the gravitational forces (F_gc and F_gt) due to the slope of the hill.

Step-by-step solution

Analyze Forces on Level Ground

In the given scenario, when the car and trailer are moving at a constant velocity on level ground, we know that the forces are balanced as there is no acceleration. The force on the car due to the trailer is equal to the force on the trailer due to the car, \(F_{\mathrm{t}}=F_{\mathrm{c}}\).

Analyze Forces when Accelerating up a Hill

When the car and trailer are accelerating up a hill, the situation is different. The forces are no longer balanced, as there is an acceleration. In order to find the relationship between the forces, we must consider the gravitational force acting on the car and trailer system due to the slope of the hill. Let's denote the gravitational force on the car as \(F_{gc}\) and on the trailer as \(F_{gt}\).

Consider Newton's Second Law

When the car and trailer are accelerating up the hill, we can apply Newton's second law, which states that the force (F) acting on an object is equal to the mass (\(m\)) multiplied by the acceleration (\(a\)), or \(F = ma\). We can write this equation for the car (\(C\)) and the trailer (\(T\)): \(F_c - F_{t} - F_{gc} = m_c a\) (1) \(F_t - F_{c} - F_{gt} = m_t(-a)\) (2) Where \(F_{gc}\) and \(F_{gt}\) are the gravitational forces on the car and trailer, respectively, and \(a\) is the uphill acceleration (positive for the car, negative for the trailer).

Solve the System of Equations

We can now solve the system of equations (1) and (2) for the relationship between the forces \(F_c\) and \(F_t\). Add equations (1) and (2): \(-F_{gc} - F_{gt} = (m_c-m_t)a\) (3)

Find the Relationship between \(F_c\) and \(F_t\)

We can now find the relationship between the two forces \(F_c\) and \(F_t\). Divide equation (3) by \(a\): \(- \frac{F_{gc}}{a} - \frac{F_{gt}}{a} = m_c - m_t\) The relationship between the forces on the car (\(F_c\)) and the forces on the trailer (\(F_t\)) when the car is accelerating up a hill can be affected by the masses of the car and trailer as well as the influence of the gravitational forces due to the slope of the hill.

Key concepts

Physics Force Analysis

Physics force analysis is vital in understanding how objects behave under various forces. By analyzing forces, we gain insight into the motion or rest state of the object in question. Force analysis usually involves identifying all the forces acting on the object and using Newton's laws to understand the resultant motion.

In the exercise with the car pulling a trailer on level ground, we apply force analysis to conclude that the forces on the car and the trailer must be equal when moving at a constant velocity. This is because, according to Newton's first law, an object in motion will remain in motion at the same velocity unless acted upon by an unbalanced force. Since the velocity is constant and there's no acceleration, the force the car exerts on the trailer (\(F_{c}\)) is equal to the force the trailer exerts on the car (\(F_{\text{t}}\)).

When the system starts to accelerate, such as going up a hill, force analysis requires us to account for additional forces like gravity, friction, and the force required to produce acceleration. To understand the dynamics of this accelerated system, let's delve into how constant velocity and accelerating systems differ.

Constant Velocity

Constant velocity refers to motion in which an object moves in a straight line at a consistent speed. It implies that there are no unbalanced forces acting on the object—in other words, either there are no forces at all, or all forces are balanced. This is a state of equilibrium where the object's velocity doesn't change, meaning there's no acceleration or deceleration happening.

Applying the concept of constant velocity to the exercise, we deduced that the car and trailer moving at a constant velocity across level ground indicates that all forces are balanced. It's a direct application of Newton's first law which states that an object will remain at rest or in uniform motion unless acted upon by a net external force. This understanding paves the way to analyze situations where velocity isn't constant, such as accelerating systems.

Accelerating Systems

Accelerating systems are in sharp contrast to constant velocity scenarios. Acceleration occurs when there's a change in velocity—which can happen due to an increase or decrease in speed, a change in direction, or both. In the example of the car and trailer moving up a hill, the system is accelerating because the gravitational force of the hill introduces an unbalanced force into the equation.

Newton's second law, which states that the force on an object is equal to its mass multiplied by its acceleration (\(F = ma\)), becomes essential in this case. The law provides the framework to calculate the additional force needed to overcome gravity and move the trailer uphill. By using the formula, we adjust for the slope's gravitational influence and determine the resultant force. This leads us to address the gravitational force and its critical role in this scenario.

Gravitational Force

The gravitational force is a universal force of attraction that acts between all masses. On Earth, it gives weight to physical objects and causes them to fall toward the ground when dropped. When analyzing a system involving the Earth's surface, such as a car and trailer moving up a hill, gravity plays a significant role.

In our exercise, the gravitational forces acting on both the car and the trailer (\(F_{gc}\) and \(F_{gt}\) respectively) must be considered when the system is accelerating. These forces are dependent on the mass of the objects and the steepness of the hill, contributing to the net force required for acceleration. By calculating these forces and including them in our force equations (\(F_c - F_{t} - F_{gc} = m_c a\) and \(F_t - F_{c} - F_{gt} = m_t(-a)\)), we gain a complete understanding of the relationship between the forces at play during acceleration – an understanding that is key for accurately predicting the motion of the car and trailer.