Chapter 11 — Work And Energy

Definition: Work is done when a force applied on a body causes displacement of the body in the direction of the force. It is a scalar quantity.

Mathematical expression (constant force): If a constant force F acts on a body and the body moves through a displacement s, making an angle θ between the force and displacement vectors, then
W = F s cosθ.

Special cases and sign convention:

• If θ = 0° (force parallel to displacement in same direction): W = F s (positive work).
• If θ = 90° (force perpendicular to displacement): W = 0 (no work done by that force).
• If θ = 180° (force opposite to displacement): W = −F s (negative work).

Variable force: If the force varies with position, the work done while the object moves from x1 to x2 is the area under the F(x) vs x curve: W = ∫(x1 to x2) F(x) dx.

Graphical interpretation: For a force-versus-displacement graph, the work done equals the area under the curve between the initial and final displacements. Positive area (above x-axis) gives positive work, negative area (below x-axis) gives negative work.

Units and dimensions: SI unit of work is joule (J). 1 J = 1 N·m. Dimensional formula: [M L^2 T⁻²].

Important notes:

• Work depends on the component of force along the displacement (F cosθ).
• Forces perpendicular to displacement (e.g., centripetal force in uniform circular motion) do no work and do not change kinetic energy.
• Work can be positive, negative or zero.

Quick example (numeric): A force of 10 N is applied at 30° to the horizontal and moves an object 5 m horizontally. Work = 10 × 5 × cos30° = 50 × 0.866 = 43.3 J.

What is Energy?
Energy is the capacity of a body or system to do work. Whenever a body can produce change or cause motion it possesses energy. Energy is a scalar quantity. The SI unit of energy is the joule (J). One joule = one newton-meter (1 J = 1 N·m).

Forms of Energy
Energy appears in many forms: kinetic energy (energy of motion), potential energy (stored energy due to position or configuration), thermal (heat), chemical, electrical, light (radiant), sound and nuclear energy. In mechanics we mainly deal with kinetic and potential energy.

Kinetic Energy (KE)
A body of mass m moving with speed v has kinetic energy given by KE = 1/2 m v^2. KE depends on mass and the square of speed. (SI unit: J)

Gravitational Potential Energy (GPE)
A body of mass m at height h above a chosen reference level has gravitational potential energy PE = m g h, where g is the acceleration due to gravity (≈ 9.8 m/s^2 near Earth's surface). PE depends on mass and height.

Work and Energy Relationship
Work is the process of energy transfer. For a constant force F acting through displacement s at angle θ to the direction of motion, the work done is W = F s cos θ. The work done by net force on a body changes its kinetic energy (Work–Energy Theorem): W_net = ΔKE = KE_final − KE_initial.

Conservation of Mechanical Energy
If only conservative forces (like gravity) act, mechanical energy (KE + PE) remains constant: KE_initial + PE_initial = KE_final + PE_final. In the presence of non-conservative forces (friction, air resistance), mechanical energy is not conserved; some mechanical energy is converted to other forms (e.g., heat).

Power
Power is the rate at which work is done or energy is transferred: P = W / t. SI unit is watt (W), where 1 W = 1 J/s. Electrical energy is often measured in kilowatt-hours (kWh): 1 kWh = 3.6 × 10^6 J.

Key ideas to remember
- Energy can be transformed from one form to another but total energy is conserved in an isolated system.
- Kinetic energy depends on v^2, so doubling speed quadruples KE.
- Potential energy depends on choice of reference level for height.

What is kinetic energy?
Kinetic energy is the energy possessed by an object because of its motion. Any moving object — a running child, a flowing river, a flying ball — has kinetic energy.

Formula
For a body of mass m moving with speed v, the kinetic energy (K or KE) is given by
KE = 1/2 m v².

SI unit: joule (J). 1 J = 1 kg·m²·s⁻².

Derivation (using work done)
If a net force F acts on a body of mass m and displaces it by s, the work done by the force is W = F·s. Using Newton's second law (F = ma) and the kinematic relation v_f² − v_i² = 2as, we get:

1. W = F s = m a s
2. Using v_f² − v_i² = 2 a s, we get W = (1/2) m (v_f² − v_i²)
3. Therefore W = ΔKE = KE_final − KE_initial. If v_i = 0, KE = (1/2) m v².

Key points to remember

• KE is a scalar quantity (no direction).
• KE ≥ 0 for any object; KE = 0 when the object is at rest relative to the chosen frame.
• KE depends on mass and the square of speed: doubling mass doubles KE; doubling speed increases KE by four times.
• Kinetic energy is frame dependent: an object may have different KE measured from different reference frames.
• Work–energy theorem: net work done on a body = change in its kinetic energy.

Energy transformations
Kinetic energy often appears by conversion from other forms: gravitational potential energy → kinetic energy (falling object), chemical energy → kinetic energy (car engine), kinetic energy → thermal energy (brakes).

What is Potential Energy?
Potential energy (PE) is the energy possessed by an object because of its position or configuration. It is stored energy that can be converted into other forms (for example, kinetic energy) when conditions change.

Types of Potential Energy relevant for Class 9

• Gravitational potential energy (near Earth's surface): Energy due to an object's height above a chosen reference level. If an object of mass m is raised by height h, the increase in its gravitational potential energy is U = mgh (where g is acceleration due to gravity ≈ 9.8 m/s²).
• Elastic potential energy: Energy stored in a stretched or compressed spring or elastic object. For a spring that obeys Hooke's law (force F = kx), the elastic potential energy stored when displaced by x is U = 1/2 k x².
• Chemical potential energy: Energy stored in chemical bonds (e.g., food, fuels, batteries). This is released or absorbed in chemical reactions.

Why mgh? (Short derivation)
To lift a mass m steadily upward through height h against gravity we do work W = force × distance = mg × h. This work goes into increasing the object’s gravitational potential energy, so ΔU = mgh.

Reference level and sign
Potential energy depends on the chosen zero (reference) level. Only changes in potential energy are physically meaningful. For gravitational PE near Earth we often set U = 0 at ground level, but any reference is allowed.

Conservation of Mechanical Energy (simple statement)
In absence of non-conservative forces (like friction), the total mechanical energy E = KE + PE remains constant. As PE decreases, KE increases and vice versa (example: a falling object converts gravitational PE to kinetic energy).

Units and dimensions
Unit of potential energy is the joule (J). 1 J = 1 N·m. Dimension: [M L² T⁻²].

Important notes for Class 9

• Potential energy is stored energy due to position/configuration.
• Only changes in potential energy matter for work and energy conversion.
• Gravitational PE near Earth is linear in height (mgh); elastic PE is quadratic in displacement (1/2 k x²).

Statement: The net work done on an object by all forces acting on it is equal to the change in its kinetic energy.

Meaning: If an object of mass m has initial speed u and final speed v after some displacement, then the total (net) work W_net done on it satisfies W_net = K_final - K_initial, where kinetic energy K = 1/2 m v².

Derivation (brief): Take motion along the line of action of the net force. By Newton's second law, F_net = m a. A small amount of work done when the object moves a small displacement ds is dW = F_net · ds. Using a = dv/dt and ds = v dt, dW = m a v dt = m v dv. Integrating from initial speed u to final speed v gives

W_net = ∫_u^v m v dv = 1/2 m v² - 1/2 m u² = ΔK.

Important points:

• W_net means the algebraic sum of work done by all forces (positive for forces that increase speed, negative for forces that decrease speed).
• If W_net = 0 then the speed does not change and kinetic energy remains constant.
• The theorem assumes constant mass and applies to the net force (vector sum). For non-collinear cases use vector dot product F · ds and integrate along the path.
• The work–energy theorem is consistent with conservation of mechanical energy: when only conservative forces act, work is converted between kinetic and potential energy; non-conservative forces (like friction) do work that changes the total mechanical energy.

What is mechanical energy? Mechanical energy of a system is the sum of its kinetic energy (energy of motion) and potential energy (energy due to position). For common Class 9 problems, KE = 1/2 m v² and gravitational PE = m g h.

Statement (Conservation of Mechanical Energy): If only conservative forces (like gravity or ideal spring force) do work on a system and non-conservative forces (like friction or air resistance) are absent or negligible, the total mechanical energy remains constant. In other words, the sum of kinetic and potential energies at any two instants is equal:

KE₁ + PE₁ = KE₂ + PE₂

Simple reasoning / derivation (qualitative): When an object falls under gravity, its potential energy decreases and that lost potential energy appears as increased kinetic energy. If no energy is lost to heat or sound, the loss in PE exactly equals the gain in KE, so the total stays the same.

Example relation from energy conservation: For an object of mass m released from rest at height h₁ and reaching height h₂ with speed v,

m g h₁ + 0 = m g h₂ + 1/2 m v²

so 1/2 m v² = m g (h₁ − h₂). If released from rest and h₂ = 0, v = sqrt(2 g h₁).

When mechanical energy is not conserved: If friction or air resistance is present, mechanical energy decreases; some mechanical energy is transformed into thermal energy, sound, etc. In that case, work done by non-conservative forces = change in mechanical energy (W_nc = Δ(KE + PE)).

Including elastic potential: For systems with springs, elastic potential energy U_s = 1/2 k x² is included in total mechanical energy: KE + m g h + 1/2 k x² = constant (when no non-conservative forces act).

Definition: Power is the rate at which work is done or the rate at which energy is transferred. It tells how fast work is being done.

Average power over a time interval Δt is given by P = W / Δt, where W is the work done in that interval. The SI unit of power is the watt (W): 1 W = 1 J/s. Another common unit is horsepower (hp): 1 hp ≈ 746 W.

Instantaneous power: If work changes continuously, instantaneous power is the time derivative of work: P = dW/dt. Using dW = F · ds for a small displacement ds under force F, dividing by dt gives the important relation for a moving object:

P = F · v (dot product). In scalar form, when force and velocity are along the same line, P = F v. If force and velocity are perpendicular, P = 0 (force does no work on the object at that instant). A negative power means the force is taking energy out of the system (opposes motion).

Useful forms: when lifting a mass m through height h in time t, work W = m g h and average power P = m g h / t. When a constant force F moves an object at constant speed v, P = F v.

What is work? Work done by a force on an object is the product of the component of the force along the displacement and the magnitude of that displacement. For a constant force F making an angle θ with the displacement s:

W = F s cosθ

Sign of work:

• W > 0 (positive) when the component of the force is in the same direction as displacement (0 < θ < 90°).
• W = 0 when the force is perpendicular to displacement (θ = 90°) or when there is no displacement.
• W < 0 (negative) when the force has a component opposite the displacement (90° < θ < 180°).

Special cases (common forces):

• Gravitational force (weight): If an object of mass m is moved vertically by height h, the work done by gravity is W = -m g h if the object is lifted up (gravity opposes displacement). If the object moves down by h, gravity does positive work +m g h. Importantly, work done by gravity depends only on vertical displacement (height change), not the path.
• Normal force: Normal is usually perpendicular to the surface. For a displacement along the surface, normal force does zero work because it is perpendicular to displacement.
• Frictional force: Friction opposes motion, so it does negative work. If kinetic friction f_k acts opposite a displacement s along the surface, W_friction = -f_k s.
• Spring force (Hooke's law): For a spring with constant k, force F = -k x (x measured from natural length). Work done by the spring when it is stretched/compressed from x = 0 to x = X is W_spring = 1/2 k X^2 (the spring does negative work on the agent stretching it but positive work on the mass if it is released—sign depends on chosen direction).
• Variable forces: When force varies with position, work is the area under the force–displacement curve: W = ∫ F · ds (from initial to final position).
• Work–energy theorem: The net work done on an object equals its change in kinetic energy: W_net = ΔK = 1/2 m v_f^2 − 1/2 m v_i^2.

Key ideas to remember: Work depends on the component of the force along the displacement, not on the total force. Forces perpendicular to motion (like centripetal or normal on flat motion) do no work. Path independence applies for conservative forces like gravity and spring force (work depends only on initial and final positions), but friction is non-conservative (work depends on the path length).

What are units and dimensions?
Units are standard measures used to express physical quantities (for example, metre for length, kilogram for mass, second for time). Dimensions describe the physical nature of a quantity in terms of basic physical quantities (for example, the dimension of velocity is length/time).

Base quantities and SI units
The seven SI base quantities commonly used are: length (L) — metre (m), mass (M) — kilogram (kg), time (T) — second (s), electric current — ampere (A), temperature — kelvin (K), amount of substance — mole (mol), luminous intensity — candela (cd). In mechanics we usually work with M, L and T.

Dimensional formula and notation
The dimensional formula of a quantity expresses it as powers of base dimensions M, L, T (and others if needed). We write it as M^a L^b T^c. Example: velocity v = distance/time so Dim(v) = L T^-1. The principle of dimensional homogeneity states that all terms in a physical equation must have the same dimensions.

Why dimensional analysis is useful
- Check correctness of formulas (dimensions must match).
- Obtain the form of a relationship up to a dimensionless constant.
- Convert units and derive units of derived quantities.
Limitations: it cannot give numerical dimensionless constants, signs, or dimensionless functions (like trigonometric or exponential dependence).

Common derived quantities (dimension & SI unit)

• Displacement/Length: Dim = L ; unit = m
• Time: Dim = T ; unit = s
• Velocity: Dim = L T^-1 ; unit = m s^-1
• Acceleration: Dim = L T^-2 ; unit = m s^-2
• Force: Dim = M L T^-2 ; unit = newton (N) = kg m s^-2
• Work / Energy: Dim = M L² T^-2 ; unit = joule (J) = N m = kg m² s^-2
• Power: Dim = M L² T^-3 ; unit = watt (W) = J s^-1
• Pressure: Dim = M L^-1 T^-2 ; unit = pascal (Pa) = N m^-2

How to find dimensional formula (example)
Given W = F × s. Dim(F) = M L T^-2, Dim(s) = L. So Dim(W) = (M L T^-2) (L) = M L² T^-2.

Overview

Applications and Numerical Problems in Work and Energy cover how work and energy concepts are used to analyse real situations and solve quantitative questions. Main ideas used are: work (scalar product of force and displacement), kinetic energy, potential energy (gravitational and elastic), work–energy theorem, conservation of mechanical energy (when non‑conservative forces are absent), and power.

Key ideas and problem strategy

• Identify forces doing work and the path of motion. Choose sign convention (usually displacement direction positive).
• If force is constant and along displacement: W = F d cosθ. If force varies: W = area under F vs x curve or W = ∫F dx.
• Use work–energy theorem: net work by all forces = change in kinetic energy, W_net = ΔK = 1/2 m(v_f^2 − v_i^2).
• Use conservation of mechanical energy when only conservative forces (gravity, spring) act: K_i + U_i = K_f + U_f. If non‑conservative forces (friction) do work, include W_nc: K_i + U_i + W_nc = K_f + U_f.
• For springs: potential energy U_s = 1/2 k x^2. Work done by spring when stretched/compressed: W_spring = −ΔU_s = −(1/2 k x_f^2 − 1/2 k x_i^2).
• Power: average P = W/Δt. Instantaneous P = F·v.

Sample numerical problems (solved)

1. Constant force along displacement
   Problem: A horizontal force of 10 N pulls a box 5 m along the same direction. Work done?
   Solution: W = F d = 10 × 5 = 50 J.
2. Lifting against gravity (constant speed)
   Problem: A 5 kg weight is lifted vertically by 2.0 m at constant speed. Find work done by (a) gravity, (b) the person doing the lifting. Use g = 9.8 m/s2.
   Solution: Change in gravitational potential ΔU = m g h = 5 × 9.8 × 2 = 98 J. Work done by gravity = −98 J (gravity acts downward while displacement is upward). Work done by person = +98 J (to raise at constant speed net work on mass is zero, so applied force equals weight).
3. Work–energy theorem
   Problem: A car of mass 1000 kg speeds up from 10 m/s to 20 m/s. Net work done on car?
   Solution: ΔK = 1/2 m(v_f^2 − v_i^2) = 0.5 × 1000 × (400 − 100) = 0.5 × 1000 × 300 = 150000 J = 1.5 × 10^5 J.
4. Spring energy
   Problem: A spring with k = 200 N/m is compressed by 0.10 m. Energy stored?
   Solution: U = 1/2 k x^2 = 0.5 × 200 × (0.10)^2 = 0.5 × 200 × 0.01 = 1.0 J.
5. Inclined plane with friction (using energy)
   Problem: A 2 kg block is pushed up a rough plane to height 1.5 m. If friction does 12 J of work (negative), how much work must the push supply to raise it at constant speed?
   Solution: Required increase in potential = m g h = 2 × 9.8 × 1.5 = 29.4 J. Work by friction = −12 J. Net work needed to change energy = ΔU + work_friction = 29.4 + 12 = 41.4 J. So applied force must do +41.4 J.

Tips for numerical problems

• Draw free‑body diagram and energy bar (K and U) before calculations.
• Check whether to use work formula directly, work–energy theorem, or energy conservation — energy methods often simplify problems with variable forces or pathways.
• Use area under F vs x graph for variable forces: area = work.
• Watch signs: work done by conservative field decreases its potential (work by gravity is negative when object is raised).