Chapter 7: Formula Reference — Mechanics
Free study material · concepts, shortcuts & solved questions
This chapter, and the chapters that follow, provide a large, well-organised reference of important formulas used across the branches of physics tested in SSC and RRB general science sections. Every formula is accompanied by a definition of the symbols used and the SI unit of the quantity concerned, so that the same table can be used both for learning the physics and for quick pre-exam revision.
7.1 Motion (Kinematics)
Kinematics is the study of the motion of objects without reference to the forces that cause the motion. The three equations of motion below apply only to motion with uniform (constant) acceleration, and are among the most heavily tested numerical formulas in SSC/RRB exams, frequently appearing in problems involving freely falling bodies, where the constant acceleration is simply the acceleration due to gravity, g.
Quantity / Law | Formula | Symbols & SI Units |
|---|---|---|
Average speed | speed = distance / time | distance in m, time in s, speed in m/s |
Average velocity | v = displacement / time | v in m/s (vector) |
Equation of motion 1 | v = u + at | u = initial velocity, v = final velocity (m/s), a = acceleration (m/s²), t = time (s) |
Equation of motion 2 | s = ut + ½at² | s = displacement (m) |
Equation of motion 3 | v² = u² + 2as | — |
Distance in nth second | sₙ = u + a(2n − 1)/2 | sₙ = distance travelled in nth second (m) |
Acceleration due to gravity (equations) | v = u ± gt; s = ut ± ½gt² | g ≈ 9.8 m/s² (downward taken +, upward taken −, or vice versa by convention) |
7.2 Force and Newton's Laws of Motion
Sir Isaac Newton's three laws of motion, published in 1687 in his landmark work Principia Mathematica, form the foundation of classical mechanics. The first law (the law of inertia) states that a body remains at rest or in uniform motion in a straight line unless acted upon by an external unbalanced force. The second law quantifies force as the rate of change of momentum, giving the familiar relation F = ma. The third law states that for every action there is an equal and opposite reaction. The formulas below follow directly from these three laws.
Quantity / Law | Formula | Symbols & SI Units |
|---|---|---|
Newton's Second Law | F = ma | F = force (N), m = mass (kg), a = acceleration (m/s²) |
Weight | W = mg | W = weight (N), g = acceleration due to gravity (m/s²) |
Momentum | p = mv | p = momentum (kg·m/s), v = velocity (m/s) |
Newton's Second Law (impulse form) | F = Δp/Δt | Δp = change in momentum, Δt = time interval |
Impulse | J = FΔt = Δp | J = impulse (N·s), same unit as momentum |
Law of Conservation of Momentum | m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | For an isolated system with no external force |
Frictional force | f = μN | μ = coefficient of friction (dimensionless), N = normal reaction (N) |
7.3 Work, Energy and Power
Work is said to be done whenever a force produces a displacement in the body on which it acts, and energy is defined as the capacity of a body to do work. Power measures how quickly work is done or energy is transferred. These three related quantities, together with the law of conservation of energy, form one of the most important and frequently tested sections of mechanics.
Quantity / Law | Formula | Symbols & SI Units |
|---|---|---|
Work done | W = F·s·cosθ | F = force (N), s = displacement (m), θ = angle between F and s |
Kinetic energy | KE = ½mv² | m = mass (kg), v = velocity (m/s); KE in joule (J) |
Potential energy (gravitational) | PE = mgh | h = height above reference (m); PE in joule (J) |
Elastic potential energy (spring) | PE = ½kx² | k = spring/force constant (N/m), x = extension/compression (m) |
Work–Energy Theorem | W = ΔKE = KEf − KEi | Work done equals change in kinetic energy |
Power | P = W/t = F·v | P = power (W), W = work (J), t = time (s), v = velocity (m/s) |
Law of Conservation of Energy | Total energy of an isolated system remains constant | Energy only changes form, is neither created nor destroyed |
Commercial unit of energy | 1 kWh (unit) = 3.6 × 10⁶ J | kWh = kilowatt-hour, used for electricity billing |
7.4 Circular Motion
When a body moves along a circular path at constant speed, its velocity continuously changes direction, so it experiences an acceleration directed towards the centre of the circle, called centripetal acceleration, and correspondingly a centripetal force that keeps it moving along the curved path. Centrifugal force, often confused with centripetal force in exam questions, is not a real force at all but a pseudo-force that appears to act outward only when the motion is described from within the rotating frame of reference itself.
Quantity | Formula | Symbols & SI Units |
|---|---|---|
Angular velocity | ω = θ/t = 2π/T = 2πf | ω in rad/s, T = time period (s), f = frequency (Hz) |
Linear (tangential) velocity | v = rω | r = radius of circular path (m) |
Centripetal acceleration | a = v²/r = ω²r | a in m/s², directed towards the centre |
Centripetal force | F = mv²/r = mω²r | F in newton (N), directed towards the centre |
Centrifugal force | F = mv²/r (pseudo-force, outward, in rotating frame) | Equal in magnitude, opposite in direction to centripetal force |
7.5 Gravitation
Newton's law of universal gravitation states that every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. This single law explains not only why objects fall to the ground but also the orbits of planets around the Sun and of satellites (both natural, like the Moon, and artificial) around the Earth, as summarised by the formulas below.
Quantity / Law | Formula | Symbols & SI Units |
|---|---|---|
Newton's Law of Gravitation | F = Gm₁m₂/r² | G = 6.674 × 10⁻¹¹ N·m²/kg² (universal gravitational constant), m₁, m₂ = masses (kg), r = distance between centres (m) |
Acceleration due to gravity (surface) | g = GM/R² | M = mass of Earth (kg), R = radius of Earth (m); g ≈ 9.8 m/s² |
Variation of g with height | g' = g(1 − 2h/R) for h << R | h = height above Earth's surface (m) |
Variation of g with depth | g' = g(1 − d/R) | d = depth below Earth's surface (m) |
Escape velocity | vₑ = √(2GM/R) = √(2gR) | vₑ ≈ 11.2 km/s for Earth |
Orbital velocity (near surface) | v₀ = √(GM/R) = √(gR) | v₀ ≈ 7.9 km/s for Earth |
Kepler's Third Law | T² ∝ r³ | T = orbital period, r = mean orbital radius |
7.6 Elasticity
Quantity / Law | Formula | Symbols & SI Units |
|---|---|---|
Stress | Stress = Force / Area | SI unit: pascal (Pa) or N/m² |
Strain | Strain = Change in dimension / Original dimension | Dimensionless (no unit) |
Hooke's Law | Stress ∝ Strain (within elastic limit) | Proportionality constant = modulus of elasticity |
Young's Modulus | Y = (F/A) / (ΔL/L) | Measures elasticity in length (longitudinal stress/strain); SI unit: Pa |
Bulk Modulus | K = −(ΔP) / (ΔV/V) | Measures volume elasticity; SI unit: Pa |
Modulus of Rigidity (Shear Modulus) | η = (F/A) / θ | Measures elasticity of shape; θ = shear angle (rad); SI unit: Pa |
7.7 Fluid Mechanics
Quantity / Law | Formula | Symbols & SI Units |
|---|---|---|
Pressure | P = F/A | P in pascal (Pa), F = force (N), A = area (m²) |
Pressure due to a fluid column | P = hρg | h = height of column (m), ρ = density (kg/m³), g = 9.8 m/s² |
Pascal's Law | Pressure applied to an enclosed fluid is transmitted equally in all directions | Basis of hydraulic lift/press |
Archimedes' Principle | Buoyant force = weight of fluid displaced = Vρg | V = volume of fluid displaced (m³), ρ = fluid density (kg/m³) |
Equation of Continuity | A₁v₁ = A₂v₂ | A = cross-sectional area (m²), v = flow velocity (m/s) |
Bernoulli's Theorem | P + ½ρv² + ρgh = constant | Along a streamline, for an ideal, incompressible, non-viscous fluid |
Stokes' Law | F = 6πηrv | η = coefficient of viscosity (Pa·s), r = radius of sphere (m), v = terminal velocity (m/s) |
Surface Tension | T = F/L | T in N/m, F = force along the surface, L = length over which it acts |