Work and Energy

1. Scientific definition of work

In everyday language we say a person “works” even when standing still holding a heavy bag. In physics, work is done only when a force causes a displacement in the body on which it acts.

Definition: Work done by a force F on an object that undergoes a displacement s in the direction of the force is:

When the force and displacement are not in the same direction, the component of force along the displacement does the work. If θ is the angle between the force vector and the displacement vector:

• θ = 0° (force and displacement same direction): W = Fs (maximum positive work).
• θ = 90° (force perpendicular to displacement): W = 0 (no work done — e.g., a person carrying a bag horizontally while walking, or the Moon orbiting Earth).
• θ = 180° (force and displacement opposite): W = −Fs (negative work — e.g., friction opposing motion, brakes stopping a car).

Three conditions for work to be done:

1. A force must be applied.
2. The object must be displaced.
3. There must be a component of force in the direction of displacement (i.e., cosθ ≠ 0).

Positive work is done when force and displacement are in the same direction (e.g., pushing a block forward). Negative work is done when force and displacement are opposite (e.g., friction, gravity when ball is thrown upward). Zero work — even though force and displacement both exist, if they are perpendicular, no work is done (e.g., gravity on a horizontally moving object).

2. Unit of work — the Joule

The SI unit of work is the Joule (J), named after James Prescott Joule.

1 Joule is the work done when a force of 1 Newton displaces an object by 1 metre in the direction of the force.

Larger units: 1 kJ = 1000 J; 1 MJ = 10&sup6; J.

Note: Work is a scalar quantity (it has magnitude only, no direction), even though it is calculated from two vector quantities (force and displacement).

3. Energy — the capacity to do work

Energy is defined as the capacity (or ability) of an object to do work. An object has energy if it can exert a force on another object and cause displacement.

• Energy is a scalar quantity.
• SI unit: Joule (J) — same as work.
• Energy and work are interconvertible; when work is done on a system, energy is stored in it.

Forms of mechanical energy:

1. Kinetic energy (KE) — energy due to motion.
2. Potential energy (PE) — energy due to position or configuration.

Total mechanical energy E = KE + PE. In an isolated system (no friction or air resistance), E is constant.

4. Kinetic energy and its derivation

An object of mass m moving with velocity v has kinetic energy:

Derivation (from Newton’s second law and equations of motion):

Consider an object of mass m initially at rest (u = 0) on a frictionless surface. A constant force F acts on it and it moves a distance s, reaching velocity v.

Step 1 — From Newton’s second law:

Step 2 — Work done by the force:

Step 3 — From the kinematic equation v² = u² + 2as, with u = 0:

Step 4 — Substitute:

This work is stored as the kinetic energy of the object: KE = ½mv².

Key points:

• KE is always positive (mass and v² are both positive).
• KE depends on speed squared — doubling speed quadruples KE.
• KE depends on mass — a heavier object moving at the same speed has more KE.
• When a body decelerates, its KE decreases; when it accelerates, KE increases.

5. Gravitational potential energy

When an object of mass m is raised to a height h above the ground, work is done against gravity. This work is stored as gravitational potential energy:

Derivation: To lift the object at constant speed (no acceleration), the applied force must equal the weight: F = mg (upward). The work done over height h:

This equals the potential energy stored.

Key points:

• PE is measured from a reference level (usually ground). PE = 0 at ground level.
• g = 9.8 m/s² (use 10 m/s² unless told otherwise in CBSE problems).
• PE depends on height above the reference — raising an object higher increases PE.
• When an object falls, PE decreases and KE increases.
• Elastic potential energy is also stored in a stretched or compressed spring.

6. Work–energy theorem

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

Proof sketch: Using F = ma and v² = u² + 2as:

Implications:

• If net work is positive, kinetic energy increases (object speeds up).
• If net work is zero, kinetic energy remains unchanged (constant speed).
• If net work is negative, kinetic energy decreases (object slows down).

This theorem connects force (a vector) to energy (a scalar), making many problems far easier to solve.

7. Law of conservation of energy

The Law of Conservation of Energy states: Energy can neither be created nor destroyed; it can only be transformed from one form to another. The total energy of an isolated system remains constant.

Mechanical energy conservation: In the absence of friction and air resistance:

Classic example — a ball falling freely from height H:

At every point, KE + PE = mgH = constant. PE is fully converted to KE by the time the ball reaches the ground.

Pendulum: At the extreme positions (maximum height), all energy is PE; at the mean (lowest) position, all energy is KE. Energy alternates between PE and KE every half-swing.

When friction is present: Some mechanical energy converts to heat (thermal energy). Total energy (mechanical + thermal) is still conserved — but mechanical energy alone decreases. That is why on Earth, a swinging pendulum eventually stops.

Other energy conversions: Electrical energy → light + heat (bulb); chemical energy → kinetic + heat (engine); solar energy → electrical (solar cell). In every case the total energy bookkeeping always balances.

8. Power

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

Since W = Fs and s/t = v (speed), we also have:

This form P = Fv is very useful when force and speed are given directly.

SI unit of power: Watt (W), named after James Watt.

1 Watt is the power when 1 Joule of work is done in 1 second.

Other units of power:

• 1 kilowatt (kW) = 1000 W
• 1 megawatt (MW) = 10&sup6; W
• 1 horsepower (hp) = 746 W (used for engines)

Power is also a scalar quantity.

9. Average power

When the rate of doing work varies over time, we use average power:

Example: A person does 6000 J of work in 2 minutes.

Difference between power and energy:

• Energy tells us how much work can be done.
• Power tells us how fast the work is done.
• A 100 W bulb uses more energy per second than a 60 W bulb, but both can do the same total work given enough time.

10. Commercial unit of energy — the kilowatt-hour

The Joule is too small a unit for measuring household electrical energy consumption. The commercial (practical) unit used by electricity boards is the kilowatt-hour (kWh).

Definition: 1 kWh is the energy consumed by a device of power 1 kW running for 1 hour.

On your electricity bill, 1 kWh is called 1 unit. The meter in your home measures energy consumption in kWh.

Example: A 100 W bulb left on for 10 hours:

Conversion summary:

• 1 kWh = 3.6 × 10&sup6; J = 3600 kJ
• To find units consumed: Units = (Power in W × Time in h) / 1000

11. NCERT Worked Examples

12. Important solved problems and activities

13. Common mistakes and exam tips

• Work is not always done when force is applied — if there is no displacement, or force is perpendicular to displacement, W = 0.
• Negative work does not mean no work — friction, air resistance, and brakes do negative work (they remove energy from the system).
• KE is never negative (v² is always positive); PE can be negative if the reference level is above the object.
• When using conservation of energy, identify the reference level for PE carefully (usually ground = 0).
• Power is not energy — a high-power device uses more energy per second, but if used briefly, it may use less total energy than a low-power device used for a long time.
• In kWh calculations, convert Watts to kilowatts first (divide by 1000) before multiplying by hours.
• 1 kWh = 3.6 × 10&sup6; J — memorise this conversion exactly; it appears as a 1-mark fill-in frequently.
• The derivation of KE = ½mv² is a board favourite for 3-mark questions — write all four steps clearly.