Work, Energy and Power

Work is done on a body when a force applied to it produces a displacement. Work is a scalar quantity defined as the dot product of the force vector and the displacement vector.

For a constant force F acting on a body that undergoes a displacement d, making an angle θ with the force, the work done is:

W = F d cosθ

Three special cases follow directly from the cosθ factor:

• If 0° ≤ θ < 90°, cosθ is positive, so the work done is positive (e.g. gravity doing work on a falling stone).
• If θ = 90°, cosθ = 0, so the work done is zero (e.g. work done by tension on a body moving in a circle, or by gravity on a body moving horizontally).
• If 90° < θ ≤ 180°, cosθ is negative, so the work done is negative (e.g. work done by friction opposing motion, or work done by gravity on a body that is being lifted upward, since the gravitational force acts downward while the displacement is upward).

When the force is not constant — for example, the force exerted by a stretched spring, which grows with extension — the displacement is divided into a large number of infinitesimally small steps dx, over each of which the force F(x) may be treated as constant. The work done over one such small step is dW = F(x) dx, and the total work done moving from x₁ to x₂ is obtained by integrating these small contributions:

W = ∫ F(x) dx, integrated from x = x₁ to x = x₂

Graphically, if F(x) is plotted against x, this integral is exactly the area enclosed between the F-x curve and the x-axis between x₁ and x₂. This is the standard way to compute work for a variable force in board exams.

Worked Example 1 (constant force):

Given: A force of 10 N is applied on a block at an angle of 60° to the horizontal, producing a horizontal displacement of 5 m.

Formula: W = F d cosθ

Substitution: W = 10 × 5 × cos60° = 10 × 5 × 0.5

Result: W = 25 J

Worked Example 2 (variable force, area under graph):

Given: A force F(x) = (3x + 2) N acts on a particle as it moves from x = 0 to x = 2 m.

Formula: W = ∫F(x)dx from 0 to 2 = [1.5x² + 2x] from 0 to 2

Substitution: W = (1.5 × 2² + 2 × 2) − 0

Result: W = (6 + 4) J = 10 J

The kinetic energy (KE) of a body of mass m moving with speed v is the energy it possesses on account of its motion:

K = ½ m v²

Kinetic energy is a scalar, is always positive (or zero), and has the same SI unit as work, the joule.

The work-energy theorem connects work and kinetic energy: the net work done by all the forces acting on a body equals the change in its kinetic energy.

W_net = K_f − K_i = ΔK

This can be derived from Newton's second law for a constant net force F acting on mass m, producing acceleration a over displacement s, starting from speed u and ending at speed v. Using v² = u² + 2as:

F s = m a s = m × (v² − u²)/2 = ½ m v² − ½ m u² = K_f − K_i

Since F s is exactly the work done by the net force, this proves W_net = ΔK. The theorem is more general than this derivation suggests — it also holds when the force is variable and in more than one dimension, since it follows from integrating Newton's second law along the path.

The theorem is very useful because it lets us find the speed of a body after a certain displacement without working out the time taken — only the total work done is needed.

Worked Example:

Given: A body of mass 2 kg, initially at rest, is acted upon by a net force that does 100 J of work on it.

Formula: W_net = K_f − K_i, with K_i = 0 and K_f = ½ m v²

Substitution: 100 = ½ × 2 × v² ⟹ v² = 100

Result: v = 10 m/s

Potential energy (U) is the energy a body possesses by virtue of its position or configuration, rather than its motion. It is defined only for a special class of forces called conservative forces.

A force is conservative if the work it does on a body moving between two points is independent of the path taken, and depends only on the initial and final positions. Equivalently, the work done by a conservative force around any closed path is zero. Gravity and the spring force are conservative; friction and air resistance are non-conservative, because the work they do depends on the path length and they always dissipate energy as heat.

For a conservative force, potential energy is defined so that the work done by the force equals the negative of the change in potential energy:

W = −ΔU = −(U_f − U_i)

Equivalently, the force is obtained from the potential energy function as F = −dU/dx (the force points where U decreases fastest).

The most common example is gravitational potential energy near the Earth's surface. If a body of mass m is raised through a height h, the work done against gravity is stored as PE, taking the reference level (U = 0) at the starting point:

U = m g h

Only changes in potential energy are physically meaningful; the reference level (U = 0) can be chosen arbitrarily for convenience.

Worked Example:

Given: A body of mass 5 kg is raised to a height of 10 m above the ground. Take g = 9.8 m/s².

Formula: U = m g h

Substitution: U = 5 × 9.8 × 10

Result: U = 490 J

An ideal spring obeys Hooke's law: the restoring force it exerts is proportional to its displacement x from the natural length, and acts opposite to the displacement:

F = −k x

where k is the spring (force) constant (SI unit N/m), indicating the stiffness of the spring.

Since this force varies with x, the work done in stretching or compressing the spring from 0 to x is found using the area-under-the-graph method: the F-x graph is a straight line through the origin of magnitude kx, so the work done equals the area of the triangle formed:

W = ½ × base × height = ½ × x × (kx) = ½ k x²

This work is stored in the spring as elastic potential energy:

U = ½ k x²

Note U is always positive, whether the spring is stretched (x > 0) or compressed (x < 0), since x is squared.

Conservation of mechanical energy: The total mechanical energy of a system is E = K + U. When only conservative forces act (no friction, air resistance, or other dissipative forces), the total mechanical energy remains constant at every instant:

K + U = constant, i.e. K_i + U_i = K_f + U_f

This follows from the work-energy theorem together with W = −ΔU: since W_net = ΔK and W_net = −ΔU for a purely conservative system, ΔK = −ΔU, i.e. Δ(K + U) = 0.

Worked Example:

Given: A spring of force constant k = 200 N/m is compressed by x = 0.1 m and released, pushing a block of mass 0.5 kg held against it, on a frictionless surface. Find the block's speed when the spring returns to its natural length.

Formula (energy stored): U = ½ k x²

Substitution: U = ½ × 200 × (0.1)² = ½ × 200 × 0.01

Result: U = 1 J

Formula (conservation of energy): ½ m v² = U

Substitution: ½ × 0.5 × v² = 1 ⟹ v² = 4

Result: v = 2 m/s

Power measures how quickly work is done or energy is transferred. It is defined as the time rate of doing work.

The average power delivered over a time interval Δt, during which work W is done, is:

P_avg = W / Δt

The instantaneous power is the limiting value of average power as Δt → 0, i.e. the rate of doing work at a particular instant:

P = dW/dt

Since dW = F·dx = F·v dt for a force F acting on a body moving with instantaneous velocity v, instantaneous power can also be written as:

P = F·v = F v cosθ

where θ is the angle between the force and velocity vectors. Power is a scalar quantity; its SI unit is the watt (W), where 1 W = 1 J/s.

A commonly used unit is the kilowatt-hour (kWh), which is actually a unit of energy (not power) — the energy consumed at the rate of 1 kW for one hour. This is the unit electricity bills are calculated in.

1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J

Another traditional (non-SI) unit of power is the horsepower: 1 hp = 746 W.

Worked Example:

Given: A pump lifts 200 kg of water through a height of 6 m in 10 s. Take g = 9.8 m/s².

Formula: Work done = mgh; P_avg = W / t

Substitution: W = 200 × 9.8 × 6 = 11760 J; P_avg = 11760 / 10

Result: P_avg = 1176 W ≈ 1.18 kW

A collision is an event in which two or more bodies exert strong mutual forces on each other for a relatively short time. In every collision, in the absence of external forces, the total linear momentum of the system is conserved:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Collisions are classified by whether kinetic energy is also conserved:

• In an elastic collision, both momentum and total kinetic energy are conserved. No energy is converted to heat, sound, or permanent deformation. Collisions between hard billiard balls approximate this.
• In an inelastic collision, momentum is conserved but kinetic energy is not — some is lost as heat, sound, or deformation. If the bodies stick together and move with a common velocity after impact, the collision is perfectly inelastic.

Perfectly inelastic collision: for masses m₁, m₂ with initial velocities u₁, u₂ along the same line, sticking together after impact, momentum conservation gives the common final velocity v:

v = (m₁u₁ + m₂u₂) / (m₁ + m₂)

Elastic collision (one dimension): solving momentum and KE conservation simultaneously for masses m₁, m₂ with initial velocities u₁, u₂ gives:

v₁ = [(m₁ − m₂)u₁ + 2m₂u₂] / (m₁ + m₂)

v₂ = [(m₂ − m₁)u₂ + 2m₁u₁] / (m₁ + m₂)

Two useful special cases with body 2 initially at rest (u₂ = 0):

• If m₁ = m₂: v₁ = 0 and v₂ = u₁ — the bodies exchange velocities.
• If m₁ ≪ m₂ (light body strikes a very heavy stationary body): v₁ ≈ −u₁ (bounces back) and v₂ ≈ 0.

Worked Example (perfectly inelastic collision):

Given: A body of mass 2 kg moving at 5 m/s collides and sticks to a stationary body of mass 3 kg.

Formula: v = (m₁u₁ + m₂u₂) / (m₁ + m₂)

Substitution: v = (2 × 5 + 3 × 0) / (2 + 3) = 10 / 5

Result: v = 2 m/s

Loss in kinetic energy: K_i = ½ × 2 × 5² = 25 J; K_f = ½ × 5 × 2² = 10 J. Energy lost = 25 − 10 = 15 J, converted to heat/sound/deformation.