Problem 48 - Algebra and Trigonometry

Question

Work A car drives 500 \(\mathrm{ft}\) on a road that is inclined \(12^{\circ}\) to the horizontal, as shown in the following figure. The car weighs 2500 Ib. Thus gravity acts straight down on the car with a constant force \(\mathbf{F}=-2500 \mathbf{j}\) . Find the work done by the car in overcoming gravity.

Step-by-Step Solution

Answer

The work done by the car is approximately 260,000 ft-lb.

Step 1: Understanding Work Done Against Gravity

The work done by a force is calculated by the formula \( W = F \, d \, \cos(\theta) \), where \( F \) is the force, \( d \) is the distance moved in the direction of the force, and \( \theta \) is the angle between the force and direction of movement. Here, the force due to gravity is \( 2500 \, \text{lb} \). The distance is 500 ft. \( \theta \) is the angle of the incline, which is 12°. We will need to find the component of gravitational force parallel to the incline.

Step 2: Calculating Component of Force Parallel to Incline

The component of gravitational force acting parallel to the incline is given by \( F_{\text{parallel}} = F \, \sin(\theta) \). Substituting the known values, \( F_{\text{parallel}} = 2500 \, \text{lb} \times \sin(12^\circ) \). Calculate \( \sin(12^\circ) \approx 0.2079 \), thus \( F_{\text{parallel}} \approx 520 \text{ lb} \).

Step 3: Calculating Work Done in Overcoming Gravity

Using the work done formula, substitute \( F_{\text{parallel}} \approx 520 \, \text{lb} \) and \( d = 500 \, \text{ft} \), so \( W = F_{\text{parallel}} \cdot d = 520 \, \text{lb} \times 500 \, \text{ft} \). Calculate \( W \approx 260000 \, \text{ft-lb} \).

Step 4: Finalizing the Answer

The work done by the car in overcoming gravity as it moves up the incline is approximately \( 260000 \, \text{ft-lb} \). This value represents the energy exerted by the car against the gravitational force while moving up.

Key Concepts

Inclined Plane

An inclined plane is a flat surface that is tilted at an angle to the horizontal. It is a crucial concept in physics and engineering when analyzing motion and forces. When a car moves on an inclined plane, the forces acting on the car include gravity, friction, and any external forces applied.

• Gravity works downward, perpendicular to the horizontal plane.
• External forces, like the car's engine power, work along the surface of the incline.

Understanding how these forces interact on an incline helps us calculate work, which is energy transferred by a force. For example, when the car moves up the incline, it has to work against both gravitational force and any friction present. This concept is part of a larger study within mechanics, which investigates the motion of bodies and the forces acting upon them.

Gravitational Force

Gravitational force is a fundamental force that attracts two bodies towards each other. It acts on objects whenever they have mass and are in the vicinity of another massive body, such as Earth.
In our scenario, the gravitational force pulling the car downwards is represented by the equation \[ F = mg \]where \( m \) is the mass and \( g \) is the acceleration due to gravity. When a car moves up an inclined road, only a component of this gravitational force acts parallel to the plane.
To find this component, we use the formula:\[ F_{\text{parallel}} = F \sin(\theta) \]This results in the calculation of how much of the gravitational force is acting along the incline, which is critical for determining the work done in overcoming it. This component is essential since it tells us how hard gravity is working to pull the car back down the slope, which the engine must counteract.

Work-Energy Principle

The work-energy principle states that the work done by the sum of all forces acting on an object equals the change in kinetic energy of that object. This principle is essential in solving problems involving movement and forces, such as those encountered with an inclined plane.

• In this exercise, the work done by the car is calculated by the formula: \[ W = F_{\text{parallel}} \cdot d \]
• This work calculates how much energy is needed to move the car up the inclined plane against gravity.

By comprehending the amount of work, you understand the energy required to counteract the gravitational pull while moving up an incline. This insight is valuable for evaluating efficiency in mechanical systems and is widespread in both theoretical physics and practical engineering problems.