Work, Energy and Power: NEET notes

1. The scalar product

NCERT § "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).

2. Work

NCERT § "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.

3. Kinetic energy and the work-energy theorem

NCERT § "Notions of Work and Kinetic Energy: 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.

4. Work done by a variable force

NCERT § "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.

5. Potential energy and conservative forces

NCERT § "The Concept of Potential Energy"

• 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.

6. Conservation of mechanical energy

NCERT § "The 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.

7. Potential energy of a spring

NCERT § "The 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.

8. Power

NCERT § "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.

9. Collisions

NCERT § "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.

Must-know facts

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

Common traps

Formulas

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.