Energy and Work

Energy and work are fundamental concepts in physics that help us understand how objects interact and how changes occur in physical systems. Energy is the capacity to do work, and work is the transfer of energy that occurs when a force moves an object over a distance.

These concepts are closely related to each other and to the concept of power, which is the rate at which work is done or energy is transferred.

Work is done when a force causes an object to move in the direction of the force. In physics, work has a specific definition:

When a constant force F acts on an object while it moves a distance d in the direction of the force, the work done is:

W = F × d

If the force and displacement are not in the same direction, we use the component of the force in the direction of the displacement:

W = F × d × cos(θ)

Where θ is the angle between the force and displacement vectors.

The SI unit of work is the joule (J), which is equivalent to a newton-meter (N·m).

Important points about work:

• Work can be positive, negative, or zero
• Positive work: Force is in the same direction as displacement (0° ≤ θ < 90°)
• Negative work: Force is in the opposite direction to displacement (90° < θ ≤ 180°)
• Zero work: Force is perpendicular to displacement (θ = 90°) or there is no displacement

Energy is the capacity to do work. There are many forms of energy, but they can be broadly categorized as:

Kinetic Energy (KE): Energy of motion. For an object of mass m moving with velocity v:

KE = ½mv²

Potential Energy (PE): Energy due to position or configuration. Common types include:

Gravitational potential energy (near Earth's surface):

PE_g = mgh

Where m is mass, g is gravitational field strength, and h is height above a reference level.

Elastic potential energy (in a spring):

PE_e = ½kx²

Where k is the spring constant and x is the displacement from the equilibrium position.

Other forms of energy include:

• Thermal energy: Related to the random motion of particles
• Chemical energy: Stored in chemical bonds
• Electrical energy: Associated with electric charges
• Nuclear energy: Stored in the nucleus of atoms
• Electromagnetic energy: Carried by electromagnetic waves

The SI unit of energy is the joule (J), the same as for work.

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another or transferred from one object to another. The total energy of an isolated system remains constant.

In a closed system with no external forces doing work:

Total initial energy = Total final energy

For mechanical systems (ignoring heat and other forms of energy), this often takes the form:

KE_initial + PE_initial = KE_final + PE_final

This principle is extremely powerful for solving problems where forces may be complex or varying, but energy is conserved.

Examples of energy transformations:

• A falling object: Gravitational PE → KE
• A pendulum: Gravitational PE ↔ KE (back and forth)
• A compressed spring launching an object: Elastic PE → KE
• A car braking: KE → Thermal energy (heat)

The work-energy theorem states that the net work done on an object equals the change in its kinetic energy:

W_net = ΔKE = KE_final - KE_initial

This theorem provides a direct link between work and energy, and can be derived from Newton's Second Law and the definition of work.

The work-energy theorem is particularly useful when dealing with varying forces, where direct application of F = ma might be difficult.

Power is the rate at which work is done or energy is transferred:

P = W/t

Where W is work and t is time.

Power can also be expressed in terms of force and velocity:

P = F·v

Where F is force and v is velocity.

The SI unit of power is the watt (W), which is equivalent to one joule per second (J/s).

Common examples of power:

• A 60W light bulb consumes 60 joules of electrical energy every second
• A 1000W (1kW) electric motor can do 1000 joules of work per second
• A car engine might be rated at 100kW, meaning it can convert chemical energy to mechanical energy at a rate of 100,000 joules per second

Power is an important consideration in practical applications, as it tells us not just how much energy is needed, but how quickly it must be supplied or converted.