Important Formulae and Constants

Mechanics covers the motion of bodies, forces, and energy conversions.

Equations of Motion (Uniform Acceleration)

These equations govern the linear motion of objects experiencing constant acceleration.

• First Equation of Motion: Determines final velocity (v) when initial velocity (u), acceleration (a), and time (t) are known.
  v = u + at
• Second Equation of Motion: Calculates displacement (s) over a specific duration.
  s = ut + 1/2at²
• Third Equation of Motion: Relates velocities and acceleration independent of time.
  v² = u² + 2as

Force, Momentum, and Energy

• Newton’s Second Law of Motion: States that force (F) equals the rate of change of momentum, simplified to the product of mass (m) and acceleration (a).
  F = ma
• Linear Momentum (p): The measure of a body’s motion, defined as the product of mass and velocity (v).
  p = mv
• Kinetic Energy (K.E.): The energy possessed by an object due to its motion.
  K.E. = 1/2mv² = p²/2m
• Gravitational Potential Energy (P.E.): The energy stored in an object due to its vertical position relative to a reference point, where g is acceleration due to gravity and h is height.
  P.E. = mgh
• Work Done (W): Computed as the dot product of Force (F) and Displacement (d), where θ is the angle between the force and displacement vectors.
  W = F · d · cos(θ)
• Power (P): The rate at which work is performed or energy is transferred.
  P = W/t = F · v

Gravitation and Planetary Motion

Gravitational interactions determine planetary paths, satellite velocities, and weight distribution across different celestial bodies.

Universal Law of Gravitation

Formulated by Sir Isaac Newton, this law states that every particle attracts every other particle with a force proportional to the product of their masses (m₁, m₂) and inversely proportional to the square of the distance (r) between their centers. G represents the Universal Gravitational Constant.

Acceleration Due to Gravity (g)

The acceleration experienced by a free-falling body near a celestial entity. It depends on the mass (M) and radius (R) of the planet or moon.

Orbital and Escape Velocity

• Orbital Velocity (v_o): The speed required for a satellite to remain in a stable circular orbit around a planet.
  v_o = √(GM/R) = √(gr)
• Escape Velocity (v_e): The minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body permanently.
  v_e = √(2GM/R) = √(2) · v_o

Wave Mechanics, Optics, and Electromagnetism

These formulations describe the behavior of light, sound, mirrors, lenses, and electrical circuits.

Wave Mechanics and Sound

• Wave Velocity (v): Directly proportional to frequency (f or ν) and wavelength (λ).
  v = f λ
• Time Period (T): The inverse of frequency, representing the time taken to complete one full cycle.
  T = 1/f

Ray Optics

• Mirror Formula: Relates focal length (f), image distance (v), and object distance (u) for spherical mirrors.
  1/f = 1/v + 1/u
• Lens Formula: Establishes the relationship between focal length, image distance, and object distance for thin spherical lenses.
  1/f = 1/v – 1/u
• Refractive Index (n): The ratio of the speed of light in a vacuum (c) to the speed of light in a specific medium (v_m).
  n = c/v_m
• Power of a Lens (P): Expressed as the reciprocal of its focal length measured strictly in meters (m). The unit is Dioptres (D).
  P = 1/f (in meters)

Electricity and Magnetism

• Ohm’s Law: States that electric current (I) flowing through a conductor is directly proportional to the potential difference (V) across its ends, assuming constant physical factors like temperature. R denotes electrical resistance.
  V = IR
• Electrical Resistance (R): Dependent on the resistivity of the material (ρ), length of the conductor (l), and cross-sectional area (A).
  R = ρ l/A
• Joule’s Law of Heating: Calculates the thermal energy (H) generated by an electric current passing through a resistor over a duration (t).
  H = I^2Rt
• Electric Power (P): The rate at which electrical energy is consumed or dissipated in a circuit.
  P = VI = I^2R = V²/R

Fundamental Physics Constants

A clear understanding of absolute constants is indispensable for conceptual clarity and resolving statement-based questions in the civil services preliminary examination.

Core Conceptual Triggers for Prelims Elimination

Escape Velocity vs. Orbital Velocity

Escape velocity is always exactly √(2) times (approximately $1.414$ times) the orbital velocity of a satellite orbiting close to the surface of the planet. For Earth, the escape velocity is approximately 11.2 km/s, whereas the near-surface orbital velocity is roughly 7.92 km/s. It is independent of the mass of the escaping body.

Variation of g (Acceleration due to Gravity)

The value of g is not constant across the surface of the Earth due to the planet’s oblate spheroid shape and rotation.

• At Poles: Radius (R) is minimum, hence g reaches its maximum value.
• At Equator: Radius (R) is maximum, and centrifugal force acts outward, hence g reaches its minimum value.
• Altitude and Depth: The value of g decreases whether you go upward into the atmosphere or downward toward the core of the Earth. At the exact center of the Earth, g = 0.

Mass vs. Weight

Mass (m) is an intrinsic, fundamental property of matter that remains constant everywhere in the universe. Weight (W = mg) is a force that changes depending on the local acceleration due to gravity. An object transported to the Moon will retain its identical mass, but its weight will drop to 1/6th of its value on Earth.

Concave vs. Convex Optical Configurations

For concave mirrors, the focal length (f) is treated as negative in sign conventions, whereas for convex mirrors, it is positive. Conversely, for lenses, a convex (converging) lens possesses a positive power and positive focal length, while a concave (diverging) lens features a negative power and negative focal length.