Introduction

In the study of mechanics, two fundamental concepts-work and energy-play a central role in understanding how forces cause motion and how motion changes. Work quantifies the effect of a force acting over a distance, while energy measures the capacity of a system to perform work. The work-energy theorem connects these two ideas by stating that the net work done on an object results in a change in its kinetic energy.

Moreover, the conservation of mechanical energy principle allows us to analyze systems where energy transforms between kinetic and potential forms without loss, simplifying complex problems. These concepts are not only foundational in physics but also powerful tools in solving competitive exam problems efficiently.

This section will build your understanding from the ground up, starting with the definitions of work and energy, moving through the work-energy theorem, and culminating in the conservation of mechanical energy. Along the way, you will encounter clear explanations, diagrams, and worked examples to solidify your grasp.

Work and Energy

Work is done when a force causes an object to move. Specifically, work is defined as the product of the component of the force in the direction of displacement and the magnitude of that displacement.

Mathematically, if a constant force \(\vec{F}\) acts on an object causing a displacement \(\vec{d}\), the work done \(W\) is

Here, \(\theta\) is the angle between the force and displacement vectors. If the force is in the same direction as displacement, \(\theta = 0^\circ\) and \(\cos 0^\circ = 1\), so work is positive. If the force opposes displacement, \(\theta = 180^\circ\) and work is negative, indicating energy is taken away from the system.

Kinetic energy is the energy an object possesses due to its motion. It depends on the mass \(m\) of the object and the square of its velocity \(v\):

Potential energy is energy stored due to an object's position or configuration. The most common form is gravitational potential energy near Earth's surface, given by

Both kinetic and potential energies are scalars, meaning they have magnitude but no direction. The unit of energy and work in the metric system is the joule (J), where 1 joule equals 1 newton-meter (N·m).

Work-Energy Theorem

The work-energy theorem states that the net work done by all forces acting on a particle equals the change in its kinetic energy. This theorem provides a powerful link between force, displacement, and motion without directly solving Newton's equations.

graph TD    A[Start: Newton's Second Law: F = m a] --> B[Express acceleration as dv/dt]    B --> C[Multiply both sides by velocity v]    C --> D[Rewrite as d(1/2 m v²)/dt = F · v]    D --> E[Integrate over time to get work done]    E --> F[Net work done = Change in kinetic energy]

Derivation Outline:

Consider a particle of mass \(m\) moving under a net force \(\vec{F}\). Newton's second law gives

\(\vec{F} = m \vec{a}\), where \(\vec{a}\) is acceleration.

Acceleration can be written as the derivative of velocity \(\vec{v}\) with respect to time:

\(\vec{a} = \frac{d\vec{v}}{dt}\).

Dot-multiplied by velocity \(\vec{v}\), we get

\(\vec{F} \cdot \vec{v} = m \frac{d\vec{v}}{dt} \cdot \vec{v} = m \frac{d}{dt} \left(\frac{v^2}{2}\right)\).

Since \(\vec{v} \cdot \vec{v} = v^2\), the left side represents the instantaneous power (rate of doing work).

Integrating over the time interval from \(t_i\) to \(t_f\), or equivalently over displacement,

\[W_{net} = \int \vec{F} \cdot d\vec{r} = \int m \frac{d}{dt} \left(\frac{v^2}{2}\right) dt = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 = \Delta K\]

This shows that the net work done on the particle equals the change in its kinetic energy.

The work-energy theorem is especially useful when forces vary or when acceleration is not constant, allowing us to bypass complicated force and acceleration calculations.

Conservation of Mechanical Energy

In many physical systems, forces can be classified as conservative or non-conservative.

• Conservative forces (like gravity and spring force) have the property that the work done by them depends only on the initial and final positions, not on the path taken. They store energy as potential energy.
• Non-conservative forces (like friction and air resistance) dissipate mechanical energy, usually converting it into heat or sound.

When only conservative forces do work, the total mechanical energy (sum of kinetic and potential energies) of the system remains constant:

This principle allows us to analyze motion by equating energies at different points, avoiding direct force calculations.

In the pendulum shown, at the highest point, the bob has maximum potential energy and zero kinetic energy (momentarily at rest). At the lowest point, potential energy is minimum, and kinetic energy is maximum. The total mechanical energy remains constant if air resistance and friction at the pivot are negligible.

When non-conservative forces like friction act, mechanical energy decreases, and the conservation principle no longer holds exactly. Instead, the work-energy theorem including non-conservative work must be used.

Worked Example 1: Using Work-Energy Theorem to Find Velocity

Worked Example 2: Calculating Work Done by a Variable Force

Worked Example 3: Energy Conservation with Friction

Worked Example 4: Potential Energy Change in a Spring System

Worked Example 5: Work Done by Gravity in Orbital Motion