Work, Energy and Power: NEET notes.

01 · NCERT § "The Scalar Product"

The scalar product

• The scalar (dot) product of two vectors is A·B = AB cos θ, with θ the angle between them; the result is a scalar.
• In components, A·B = AxBx + AyBy + AzBz; for unit vectors î·î = 1 and î·ĵ = 0.
• The dot product is commutative (A·B = B·A) and distributive over addition.
• A·B is positive for θ < 90°, zero for θ = 90° and negative for θ > 90°.
• Geometrically, A·B is the magnitude of one vector times the projection of the other on it.
• The angle between two vectors can be found from cos θ = A·B/(AB).

02 · NCERT § "Work"

Work

• Work done by a constant force F during a displacement d is W = F·d = Fd cos θ. Only the component of force along the displacement does work.
• Work is a scalar; its SI unit is the joule, 1 J = 1 N m. Other units: 1 erg = 10⁻⁷ J, 1 eV = 1.6 × 10⁻¹⁹ J, 1 kWh = 3.6 × 10⁶ J.
• No work is done if the displacement is zero (pushing a rigid wall), if the force is zero (a body moving freely at constant velocity), or if force and displacement are perpendicular.
• Work can be negative: the force of friction on a sliding body and the gravitational force on a rising body both do negative work.
• Always name the force whose work is asked for: the work done by gravity, by friction and by the applied force on the same body are separate quantities.
• When several forces act, the net work is the sum of the work done by each, which also equals the work done by the net force.

03 · NCERT § "Notions of Work and Kinetic Energy: The Work-Energy Theorem"

Kinetic energy and the work-energy theorem

• Kinetic energy of a body of mass m moving at speed v is K = ½mv², a non-negative scalar.
• In terms of momentum, K = p²/2m; so at equal momentum the lighter body has more kinetic energy.
• Work-energy theorem: the net work done on a body by all forces equals the change in its kinetic energy, K_f − K_i = W_net.
• The theorem follows from the second law combined with the kinematics, and it is a scalar statement, so directions enter only through the signs of the work terms.
• Doubling speed makes kinetic energy four times larger; a 30% rise in momentum raises kinetic energy by 69% since K ∝ p² for fixed mass.
• A body sliding to rest on a rough floor loses all its kinetic energy to the negative work of friction: ½mv² = μ_k m g d, so the stopping distance d = v²/(2μ_k g) does not depend on mass.

04 · NCERT § "Work Done by a Variable Force"

Work done by a variable force

• When a force varies with position, split the displacement into small steps and add F(x)Δx; in the limit, W = ∫F(x) dx between the initial and final positions.
• Graphically, the work done is the area under the force-displacement graph, taken with sign.
• The work-energy theorem holds for variable forces as well; it is proved by integrating the second law over the displacement.
• Work by a spring from x_i to x_f is −½k(x_f² − x_i²), which follows from integrating F = −kx.
• Area below the displacement axis on an F-x graph represents negative work.

05 · NCERT § "The Concept of Potential Energy"

Potential energy and conservative forces

• Potential energy is energy stored by virtue of a body's position or configuration. It is defined only for conservative forces.
• A force is conservative if the work it does depends only on the start and end points, not on the path; equivalently, the work it does over any closed path is zero.
• Gravity near the earth's surface and the spring force are conservative. Friction and air drag are not, since the work they do depends on the path.
• For a conservative force in one dimension, F(x) = −dV/dx: the force points towards decreasing potential energy.
• Near the earth's surface the gravitational potential energy of a mass m at height h is V(h) = mgh, measured from a chosen zero level.
• Only changes in potential energy have physical meaning; the zero level can be chosen wherever convenient.
• Potential energy has the same unit as work and kinetic energy, the joule.

06 · NCERT § "The Conservation of Mechanical Energy"

Conservation of mechanical energy

• If only conservative forces do work on a body, its total mechanical energy K + V stays constant.
• A body falling freely from height H has the same total energy mgH at every point: potential energy turns into kinetic energy as it falls.
• Speed at the bottom of a smooth curved track depends only on the height dropped, v = √(2gh), not on the shape of the track.
• For a bob whirled in a vertical circle on a string of length L, the minimum speed at the lowest point for completing the circle is √(5gL), and the minimum speed at the top is √(gL).
• When non-conservative forces like friction act, the loss in mechanical energy equals the work done against them: (K + V)_initial − (K + V)_final = work done against friction.
• The energy lost to friction is not destroyed; it appears mainly as heat, so total energy is still conserved.

07 · NCERT § "The Potential Energy of a Spring"

Potential energy of a spring

• An ideal spring obeys Hooke's law, F_s = −kx, where x is the extension or compression from the natural length and k is the spring constant (N m⁻¹).
• The potential energy stored in a spring deformed by x is V(x) = ½kx², the same for extension and compression by the same amount.
• A stiffer spring (larger k) stores more energy for the same deformation.
• On a smooth surface, a block of speed v pushing a spring comes to rest at maximum compression x_m where ½mv² = ½kx_m², so x_m = v√(m/k).
• Energy in a mass-spring system shifts between kinetic and potential forms; the kinetic energy is greatest at the natural length and zero at maximum deformation.
• The work done by the spring force over a complete cycle (returning to the starting deformation) is zero, as expected for a conservative force.

08 · NCERT § "Power"

Power

• Power is the rate of doing work. Average power = W/t; instantaneous power P = dW/dt.
• Instantaneous power can also be written P = F·v, the dot product of force and velocity.
• The SI unit of power is the watt, 1 W = 1 J s⁻¹. One horsepower is 746 W.
• The kilowatt hour is a unit of energy, not power: 1 kWh = 3.6 × 10⁶ J.
• A pump lifting mass m of water through height h in time t does useful work at the rate mgh/t; if its efficiency is η, the input power is (mgh/t)/η.
• Efficiency is useful output power divided by input power and is always less than 1 for real machines.

09 · NCERT § "Collisions"

Collisions

• In every collision the total linear momentum of the colliding bodies is conserved, because the forces between them are internal and equal and opposite.
• In an elastic collision total kinetic energy is also conserved; in an inelastic collision part of it becomes heat, sound or deformation.
• In a completely inelastic collision the bodies stick together after impact and the loss of kinetic energy is the largest possible.
• Kinetic energy is not conserved during the brief contact itself, even in an elastic collision; the statement applies to the energy before and after.
• For one-dimensional elastic collision of m₁ (speed v₁ᵢ) with m₂ at rest: v₁f = (m₁ − m₂)v₁ᵢ/(m₁ + m₂) and v₂f = 2m₁v₁ᵢ/(m₁ + m₂).
• Equal masses in a one-dimensional elastic collision exchange velocities: the moving body stops and the struck one moves off with its speed.
• A light body striking a very heavy body at rest rebounds with nearly the same speed, while the heavy body barely moves; a heavy body hitting a light one at rest carries on almost unchanged and the light one moves off at nearly twice its speed.
• For a completely inelastic collision with m₂ initially at rest, the common velocity is v = m₁v₁ᵢ/(m₁ + m₂) and the kinetic energy lost is ½[m₁m₂/(m₁ + m₂)]v₁ᵢ².
• Each body's own kinetic energy changes in an elastic collision; it is only the total that is conserved.

Revise

Must-know facts

1. 01W = F·d = Fd cos θ; zero when force and displacement are perpendicular.
2. 02Work by friction on a sliding body and by gravity on a rising body is negative.
3. 031 kWh = 3.6 × 10⁶ J; 1 eV = 1.6 × 10⁻¹⁹ J; 1 erg = 10⁻⁷ J; 1 hp = 746 W.
4. 04Work-energy theorem: W_net = ΔK, valid for constant and variable forces.
5. 05K = ½mv² = p²/2m; K ∝ p² for fixed mass.
6. 06A rise of x% in momentum raises K by ((1 + x/100)² − 1) × 100%.
7. 07Work by a variable force = area under the F-x graph.
8. 08Conservative force: path-independent work, zero work round a closed path; F = −dV/dx.
9. 09Friction and air resistance are non-conservative.
10. 10Gravitational PE near the surface = mgh; spring PE = ½kx².
11. 11Mechanical energy is conserved only when non-conservative forces do no work.
12. 12Stopping distance on a rough floor d = v²/(2μ_k g), independent of mass.
13. 13Vertical circle on a string: minimum speed √(5gL) at the bottom and √(gL) at the top.
14. 14P = dW/dt = F·v.
15. 15Momentum is conserved in all collisions; kinetic energy only in elastic ones.
16. 16Equal masses in a 1D elastic collision exchange velocities.
17. 17Fraction of KE transferred by m₁ to m₂ (at rest) in a 1D elastic collision = 4m₁m₂/(m₁ + m₂)².
18. 18Perfectly inelastic collision gives the largest KE loss; bodies move together afterwards.

Where marks go

Common traps

Reference

Formulas

Scalar product

A·B = AB cos θ = AxBx + AyBy + AzBz

θ is the angle between A and B.

Work by a constant force

W = F·d = Fd cos θ

SI unit joule (N m).

Work by a variable force

W = ∫ F(x) dx from x_i to x_f

Area under the F-x graph.

Kinetic energy

K = ½mv² = p²/2m

Always ≥ 0.

Work-energy theorem

K_f − K_i = W_net

W_net includes work by every force, conservative or not.

Conservative force from potential energy

F(x) = −dV/dx

One-dimensional form.

Gravitational potential energy near the surface

V(h) = mgh

h measured from the chosen zero level.

Conservation of mechanical energy

K_i + V_i = K_f + V_f

Only when conservative forces alone do work.

Spring force (Hooke's law)

F_s = −kx

k in N m⁻¹.

Spring potential energy

V(x) = ½kx²

Same for extension and compression x.

Work by a spring

W_s = −½k(x_f² − x_i²)

Negative when the spring is stretched further.

Power

P_av = W/t; P = dW/dt = F·v

1 W = 1 J s⁻¹; 1 hp = 746 W.

Efficiency

η = useful output power / input power

Always less than 1 in practice.

1D elastic collision, m₂ initially at rest

v₁f = (m₁ − m₂)v₁ᵢ/(m₁ + m₂); v₂f = 2m₁v₁ᵢ/(m₁ + m₂)

Momentum and total KE both conserved.

Completely inelastic collision, m₂ at rest

v = m₁v₁ᵢ/(m₁ + m₂); ΔK = ½[m₁m₂/(m₁ + m₂)]v₁ᵢ²

Bodies stick together; ΔK is the kinetic energy lost.

Vertical circle on a string

v_bottom(min) = √(5gL); v_top(min) = √(gL)

L is the string length.

Glossary

Key terms

Scalar product

A product of two vectors that gives a scalar, AB cos θ.

Work

The product of a force and the component of displacement along it.

Joule

The work done when a force of 1 N moves its point of application 1 m along the force.

Kinetic energy

The energy a body has because it is moving, ½mv².

Work-energy theorem

The net work done on a body equals the change in its kinetic energy.

Potential energy

Stored energy that depends on the position or configuration of a system.

Conservative force

A force whose work between two points does not depend on the path taken.

Non-conservative force

A force, such as friction, whose work depends on the path and which reduces mechanical energy.

Mechanical energy

Kinetic energy plus potential energy of a system.

Spring constant

The force per unit extension of a spring; a measure of its stiffness.

Power

Work done, or energy transferred, per unit time.

Elastic collision

A collision in which total kinetic energy after impact equals that before.

Inelastic collision

A collision in which some kinetic energy is converted to other forms.

Completely inelastic collision

An inelastic collision in which the bodies move together after impact.