CONSERVATION OF ENERGY | 4.1

Energy

The capacity to do work or cause change; appears in many forms (mechanical, thermal, chemical, etc.).

Work

The transfer of energy that occurs when a force causes displacement.

W=FΔxcosθ, where θ is the angle between force and displacement.

Formula for Work

Positive Work

When the force and displacement are in the same direction

Negative Work –

When the force and displacement are in opposite directions.

Zero Work –

When the force is perpendicular to the displacement.

Joule (J) = N·m = kg·m²/s².

SI Unit of Work/Energy –

Nature of Work –

Work is a scalar quantity; it has magnitude but no direction.

Total or Net Work –

The sum of work done by all forces on an object; equivalent to the work done by the net force.

Kinetic Energy (KE) – Energy of motion given by K=12mv2K = \frac{1}{2}mv^2K=21​mv2

Energy of motion given by K=1/2mv^2

Work–Energy Theorem –

The net work done on an object equals its change in kinetic energy: Wnet=ΔK

Power –

The rate at which work is done or energy is transferred.

Pavg=ΔW/Δt.

Average Power

Watt (W) = Joule per second (J/s)

SI Unit of Power

Watt (W) = Joule per second (J/s)

Potential Energy (PE)

Not a property of a single object; it depends on the interaction between objects.

Nature of Potential Energy –

Gravitational Potential Energy (GPE)

Energy due to height in a gravitational field: Ug=mghU_g = mghUg​=mgh.

Work–Potential Energy Relation

– Wgravity=−ΔUg; work by gravity is negative of potential energy change.

Elastic Potential Energy (EPE)

Energy stored in a stretched or compressed spring: Uel=1/2kx^2

positive when the spring relaxes, negative when stretched.

Work Done by a Spring

Mechanical Energy (E)

The sum of kinetic and potential energies: E=K+U

Conservation of Mechanical Energy (CME)

If only conservative forces act, total mechanical energy remains constant: K1+U1=K2+U2

Non-Conservative Forces –

Forces (like friction or air resistance) that cause mechanical energy to decrease by converting it into heat or internal energy.

Conservative Forces

Forces whose work is independent of path and fully recoverable (e.g., gravity, spring, electrostatic).

Friction, air drag, applied forces; work depends on path and converts mechanical energy to heat.

Non-Conservative Forces (Examples)

The negative of the slope (–dU/dx) represents the force at that point.

Slope of Potential Energy Graph

Stable: at a minimum of potential energy.
• Unstable: at a maximum of potential energy.
• Neutral: flat region (constant U).

Types of Equilibrium