Work, Energy and Power

In physics, the concepts of work, energy, and power are fundamental to understanding how forces interact with objects and cause changes in their motion or state. They form the basis of mechanics and have applications across all scientific disciplines.

### Work Done (W)

Work is done when a force causes an object to move through a distance. It is a measure of energy transfer. For work to be done, two conditions must be met: there must be a force acting on the object, and the object must have a displacement in the direction of the force component.

The work done by a constant force is defined as the product of the magnitude of the force and the distance moved in the direction of the force.

Formula: W = Fd

Where:

One joule is the work done when a force of one newton moves an object through a distance of one metre.

If the force is applied at an angle θ to the direction of motion, we consider only the component of the force that is parallel to the displacement.

Formula: W = Fd cos(θ)

This means if a force is perpendicular to the direction of motion (θ = 90°), no work is done (cos 90° = 0). For example, the gravitational force on a satellite in a circular orbit does no work because the force is always perpendicular to its velocity.

### Energy

Energy is defined as the capacity to do work. It exists in various forms, such as kinetic, potential, thermal, and chemical. The SI unit for energy is also the joule (J).

#### Kinetic Energy (E_k)

Kinetic energy is the energy an object possesses due to its motion. Any moving object has kinetic energy.

Formula: E_k = ½mv²

Where:

The Work-Energy Theorem states that the net work done on an object is equal to the change in its kinetic energy (ΔE_k).

#### Potential Energy (E_p)

Potential energy is the energy stored within an object due to its position, state, or arrangement.

Formula for change in GPE: ΔE_p = mgΔh

Where:

### The Principle of Conservation of Energy

This is one of the most fundamental principles in physics. It states that energy cannot be created or destroyed, but only transformed from one form to another. The total energy in a closed, isolated system remains constant.

For mechanical systems involving conservative forces (like gravity), the total mechanical energy (KE + PE) is conserved.

Initial (E_k + E_p) = Final (E_k + E_p)

However, in most real-world scenarios, non-conservative forces like friction and air resistance are present. These forces do negative work, converting mechanical energy into other forms, primarily thermal energy (heat) and sound. In such cases, the principle is applied as:

Initial Energy = Final Energy + Work done against resistive forces

For example, when a block slides down a rough slope, its initial GPE is converted into final KE *and* thermal energy due to work done against friction.

### Power (P)

Power is the rate at which work is done or the rate at which energy is transferred or transformed.

Formula: P = W/t or P = ΔE/t

Where:

One watt is equivalent to one joule per second (1 W = 1 J/s).

A useful formula for the power developed by a constant force F moving an object at a constant velocity v can be derived:

Since P = W/t and W = Fd, then P = (Fd)/t. As velocity v = d/t, we get:

Formula: P = Fv

This is particularly useful for calculating the power required by a vehicle to overcome resistive forces at a constant speed.

### Efficiency

Efficiency is a measure of how effectively energy is converted from one form to a desired useful form. No real-world process is 100% efficient; some energy is always lost to the surroundings, often as wasted heat.

Efficiency can be calculated as a ratio or a percentage:

Formula: Efficiency = (Useful energy output / Total energy input) × 100%

Since power is the rate of energy transfer, the formula can also be expressed in terms of power:

Formula: Efficiency = (Useful power output / Total power input) × 100%