Work, Power and Energy - Physics - General Awareness - Bank Exams PDF Download

Work

Whenever a force acting on a body produces a displacement of that body, we say that work is done by the force. Work is a scalar quantity and its SI unit is the joule (J), where 1 J = 1 N·m.

Work done by a constant force

If a constant force F acts on a body and the body is displaced through a vector s, the work done by the force is given by the scalar product (dot product)

W = F · s

Equivalently, if θ is the angle between the force vector and the displacement vector,

W = |F| |s| cosθ

• If θ < 90°, W is positive (force has a component along displacement).
• If θ = 90°, W = 0 (force is perpendicular to displacement and does no work).
• If θ > 90°, W is negative (force opposes the displacement).

Angle between Force and Displacement 

Examples showing sign of work

• When a body falls freely under gravity, the displacement is downward and the gravitational force is also downward (θ = 0°). The work done by gravity is positive.
• When a body is thrown up, gravity acts downward while displacement is upward (θ = 180°). The work done by gravity is negative.
• When a person carries a load horizontally at constant height, the vertical component of force supporting the load is perpendicular to the horizontal displacement (θ = 90°). The vertical support force does zero work.
• When a body is moved along a rough surface, the frictional force opposes motion. The work done by friction is negative and removes mechanical energy from the system.
• When brakes are applied to a moving vehicle, the braking force does negative work and reduces the vehicle's kinetic energy.
• When a positive charge is moved closer to another positive charge, the electrostatic force opposing motion does negative work; external work is required to bring charges closer.

Work done by a variable force

When the force varies along the path, work is obtained by integrating the infinitesimal work dW = F · dr over the path from initial point A to final point B:

W = ∫_A^B F · dr

This form is essential for central forces, springs, and any situation where force depends on position.

Work done by specific forces

• Work done by gravity (near Earth's surface): If a mass m moves vertically by height change Δh, the work done by gravity is W = -m g Δh when Δh is measured positive upward. If an object falls through height h, the work done by gravity is +m g h.
• Work done by a spring force: For a linear spring with constant k, when the spring is stretched from x = 0 to x = x, the work done by the spring force on the mass is W = -½ k x² (negative because spring force is opposite to the applied displacement direction when stretching). The work done by the external agent to stretch the spring is +½ k x².

Energy

Energy is the capacity of a body or system to do work. Energy is a scalar quantity and its SI unit is the joule (J). Energy occurs in several forms; the two most common mechanical forms are kinetic energy and potential energy.

Kinetic energy

The kinetic energy of a body of mass m moving with speed v is

K = ½ m v²

This expression can be obtained from the work-energy theorem (net work done on a particle equals the change in its kinetic energy).

Potential energy

Potential energy is energy stored in a system due to the configuration or position of bodies in a force field.

• Gravitational potential energy near Earth: For mass m at height h above a chosen reference, U = m g h. The reference level (h = 0) must be specified.
• Elastic potential energy: For a spring obeying Hooke's law (F = -k x), the potential energy stored when stretched or compressed by amount x is U = ½ k x².

Work-energy theorem

The work-energy theorem relates net work done on a particle to its change in kinetic energy:

W_net = ΔK = K_f - K_i

This theorem is derived from Newton's second law and the definition of work and is widely used to solve dynamics problems where forces and displacements are known.

Conservation of mechanical energy

When only conservative forces (such as gravity and ideal spring forces) act on a system, the total mechanical energy (kinetic + potential) remains constant:

K + U = constant

If non-conservative forces (for example, friction or air resistance) do work, the mechanical energy changes by the amount of work done by those forces:

Δ(K + U) = W_nc

where W_nc is the work done by non-conservative forces (negative when mechanical energy is dissipated).

Einstein relation between mass and energy

Mass and energy are equivalent according to special relativity. The equivalence is expressed by Einstein's mass-energy relation:

E = m c²

Here m is the mass that is converted into energy, E is the energy produced, and c is the speed of light in vacuum. Use c = 3 × 10⁸ m·s⁻¹ for numerical work in senior-level problems.

Power

Power is the rate at which work is done or the rate at which energy is transferred. Power is a scalar quantity and its SI unit is the watt (W), where 1 W = 1 J·s⁻¹.

Average and instantaneous power

The average power delivered by an agent that does work W in time t is

P_avg = W / t

The instantaneous power delivered by a force acting on a particle is

P = dW/dt = F · v

Thus, instantaneous power equals the scalar product of force and instantaneous velocity. Equivalently, if the force and velocity make angle θ,

P = |F| |v| cosθ

Units and conversions

• SI unit of power: watt (W) = J·s⁻¹.
• Horsepower: 1 horsepower (HP) = 746 W (approximately).

Applications and example problems

Work, energy and power are central in many engineering contexts:

• Civil engineering: calculation of work against gravity when lifting materials, energy methods in structural analysis, and energy dissipation in dampers.
• Electrical engineering: energy delivered by sources, power consumed by resistive loads, and conversion between electrical and mechanical power in motors and generators.
• Computer engineering and mechatronics: power budgets for actuators, energy consumption in motion control, and energy-conserving algorithms for simulations.

Sample calculation - kinetic energy gained by a block

A block of mass m is pulled by a constant horizontal force F across a frictionless surface through distance s. Find the work done by the force and the final kinetic energy if the block starts from rest.

Solution:
The work done by the force is the dot product of force and displacement.
W = F · s
Because force and displacement are collinear,
W = F s
By the work-energy theorem, the net work equals the change in kinetic energy.
ΔK = K_f - K_i = W
The block starts from rest so K_i = 0, therefore
K_f = W = F s

Sample calculation - average and instantaneous power

An electric motor lifts a load doing 1500 J of work in 10 s. Find the average power. If the motor exerts a constant upward force of 200 N and the load moves upward at a constant speed of 0.75 m·s⁻¹, find the instantaneous power delivered by the motor.

Solution:
Average power is work divided by time.
P_avg = W / t
P_avg = 1500 J / 10 s
P_avg = 150 W

Instantaneous power when force and velocity are constant and collinear is the product of force and speed.
P = F v
P = 200 N × 0.75 m·s⁻¹
P = 150 W

Summary

• Work quantifies energy transfer by a force through a displacement: W = F · s = |F||s| cosθ.
• Kinetic energy of a mass m moving at speed v is K = ½ m v².
• Potential energy depends on the configuration: gravitational U = m g h; elastic U = ½ k x².
• Work-energy theorem: net work done = change in kinetic energy.
• Conservation of mechanical energy: K + U = constant when only conservative forces act.
• Power is the rate of doing work: P = dW/dt = F · v; SI unit is watt (W); 1 HP = 746 W.
• Mass-energy equivalence: E = m c² relates mass and energy; c = 3 × 10⁸ m·s⁻¹.