Work, Energy, and Power: Work Done by Forces

Understanding Work Done by a Constant Force

In physics, work is done when a force causes a displacement. For a constant force, the work done is the product of the magnitude of the force, the magnitude of the displacement, and the cosine of the angle between the force and displacement vectors. Mathematically, this is represented as $W = Fd \cos(\theta)$ .

What are the three components needed to calculate work done by a constant force?

Magnitude of force (F), magnitude of displacement (d), and the cosine of the angle (θ) between the force and displacement vectors.

The unit of work in the SI system is the Joule (J). One Joule is defined as the work done when a force of one Newton moves an object through a distance of one meter in the direction of the force.

If the force and displacement are in the same direction (θ = 0°), work done is maximum and positive ( $W = Fd$ ). If they are in opposite directions (θ = 180°), work done is maximum and negative ( $W = -Fd$ ). If the force is perpendicular to the displacement (θ = 90°), no work is done ( $W = 0$ ).

Work Done by a Variable Force

When a force is not constant, meaning it changes in magnitude or direction (or both) during the displacement, we cannot use the simple formula $W = Fd \cos(\theta)$ . Instead, we must consider the force at each infinitesimal part of the displacement and sum up the work done over that entire path. This is achieved through integration.

Work done by a variable force is the area under the Force-Displacement graph.

For a variable force, the work done is calculated by integrating the force with respect to displacement along the path of motion. This is equivalent to finding the area under the curve on a graph where force is plotted against displacement.

Consider a force $F(x)$ that varies with position $x$ . To find the work done by this force as an object moves from position $x_1$ to $x_2$ , we divide the displacement into infinitesimally small segments, $dx$ . The work done over each segment is $dW = F(x) dx$ . The total work done is the sum of these infinitesimal works, which is given by the definite integral: $W = \int_{x_1}^{x_2} F(x) dx$ . If the force has both x and y components, and displacement is in 2D, the work done is $W = \int_{C} (F_x dx + F_y dy)$ , where C is the path of integration.

The integral $\int_{x_1}^{x_2} F(x) dx$ represents the area under the Force vs. Displacement curve between the initial position $x_1$ and the final position $x_2$ . This visual representation is crucial for understanding how to calculate work when the force is not constant. For example, if the force is a linear function of displacement, the area under the curve will be a trapezoid or a triangle, whose area can be calculated using geometric formulas.

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How is work done by a variable force calculated when the force is a function of position?

By integrating the force function with respect to displacement over the path of motion: $W = \int_{x_1}^{x_2} F(x) dx$ .

Work-Energy Theorem

A fundamental concept linking work and energy is the Work-Energy Theorem. It states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, $W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$ , where $m$ is the mass, $v_f$ is the final velocity, and $v_i$ is the initial velocity.

Feature	Constant Force	Variable Force
Calculation Method	$W = Fd \cos(\theta)$	$W = \int F \cdot ds$
Force Magnitude	Constant	Changes with position/time
Graphical Representation	Force-displacement graph is a horizontal line	Force-displacement graph is a curve
Work Calculation	Direct multiplication	Area under the F-d curve (integration)

Learning Resources

Work Done by a Constant Force - Khan Academy(video)

Learn the definition of work done by a constant force and its calculation with clear examples.

Work Done by a Variable Force - Physics Classroom(documentation)

Explains how to calculate work done by a variable force using integration and graphical methods.

Work, Energy, and Power - JEE Physics Notes(documentation)

Comprehensive notes on Work, Energy, and Power, including detailed explanations of work done by constant and variable forces relevant for JEE.

Work-Energy Theorem Explained(blog)

Understand the relationship between work done and kinetic energy through the Work-Energy Theorem.

Calculating Work Done by Variable Force - YouTube Tutorial(video)

A visual tutorial demonstrating how to calculate work done by a variable force using integration.

Work and Potential Energy - Physics LibreTexts(documentation)

A detailed chapter covering work, including work done by constant and variable forces, and the work-energy theorem.

JEE Physics: Work, Energy and Power - Practice Problems(documentation)

Provides practice problems and solutions for Work, Energy, and Power, focusing on JEE-level questions.

Work Done by a Force - Wikipedia(wikipedia)

A comprehensive overview of the physics concept of work, including its definition, units, and applications.

Understanding Work in Physics - Brilliant.org(documentation)

Interactive explanations and examples of work done by forces, including variable forces.

JEE Main 2024 Physics: Work Energy Power - Study Material(documentation)

Study material specifically curated for JEE Main, covering Work, Energy, and Power with relevant formulas and concepts.