Physics Formulas and Fundamental Laws for Students

Motion in a Plane

Magnitude of vector: |A| = √(a_x² + a_y² + a_z²)

Unit vector: â = vector a / |a|

Resultant: R = √(A² + B² + 2AB cos θ)

Time period of projectile motion (T): 2u sin θ / g

Maximum height (H): u² sin² θ / 2g

Range (R): u² sin 2θ / g

Uniform Circular Motion (UCM)

• Centripetal acceleration (a_c): v² / r
• Angular velocity (ω): θ / t

Relation between linear velocity and angular velocity: v = rω

Gravitation

The Universal Law of Gravitation: Masses attract each other with a force proportional to their masses and inversely proportional to the square of the distance between them.

Force (F): G m₁m₂ / r²

Gravity at height (g_h): g_h = g_s / (1 + h/r)² or g_h = g_s(1 - 2h/r) if h < 300 km

Gravity at depth (g_d): g_d = g_s(1 - d/R)

Escape velocity (v_e): v_e = √(2gR) = √(2GM/R)

Orbital velocity (v_o): v_o = √(GM/r) = √(gR)

Time period of an object/satellite (T): T = (2π / √GM) * r^3/2 or T = 2πr / √(GM/r)

Kepler's Laws

• Kepler's 1st Law: Orbits are ellipses with the Sun at one focus.
• Kepler's 2nd Law: Equal areas are swept out in equal times.
• Kepler's 3rd Law: The square of the time period is proportional to the cube of the semi-major axis.

Energy in Gravitation:

• Total Energy (TE): -GMm / 2r
• Kinetic Energy (KE): GMm / 2r
• Potential Energy (PE): -GMm / r

Work, Energy, and Power

Work: F s cos θ = F · s = ΔK

Kinetic Energy (KE): ½mv²

Potential Energy (PE): mgh

Power: Work / Time = Force × Velocity

Laws of Motion

• Newton's 1st Law: An object stays in motion unless acted upon by a force.
• Newton's 2nd Law: Force equals mass times acceleration (F = ma).
• Newton's 3rd Law: Interacting objects apply equal and opposite forces.

Impulse: Δp = F Δt

Recoil velocity: MV = -mv | V = -mv / M

Friction:

• Maximum static friction (f_s)_max: μ_sN
• Kinetic friction (f_k): μ_kN

Circular Motion on Roads

• Curved road (v_max): √(μ_srg)
• Banked road (v_max): √(rg tan θ) (assuming no friction)
• Banked road with friction (v_max): √[rg(μ_s + tan θ) / (1 - μ_s tan θ)] (no slipping)

Motion in a Straight Line

Kinematic Equations:

• v = u + at
• at = v - u
• a = (v - u) / t
• t = (v - u) / a
• s = ut + ½at²
• v² = u² + 2as
• a = (v² - u²) / 2s

Bernoulli's Principle: Pressure decreases as speed increases in a flowing fluid.

Hooke's Law: A spring stretches or compresses in proportion to the force applied to it.

Thermodynamics

• Zeroth Law: Bodies in thermal equilibrium with a third body are also in equilibrium with each other.
• 1st Law: Energy cannot be created or destroyed, only transformed.
• 2nd Law: In spontaneous processes, the entropy of the universe increases.
• 3rd Law: At absolute zero, a perfect crystal has zero entropy, representing the lowest possible state of disorder.

Fluids

Pressure: Force / Area

Pressure-Depth Relation: P₂ - P₁ = hρg

Gauge Pressure: P₂ - P_atm = hρg

Solids

Stress: Restoring Force (F) / Area (N/m²)

Strain: ΔD / Original dimension

Hooke's Law: Within the elastic limit, stress is proportional to strain.

Moduli of Elasticity

• Young's Modulus: Linear stress (F/A) / Linear strain (Δl/l)
• Shear Modulus: Shearing stress (F/A) / Shearing strain (θ)
• Bulk Modulus: Volume stress (F/A) / Volume strain (Δv/v)

Oscillation

Simple Harmonic Motion (SHM)

• Angular frequency (ω): √(k/m)
• Time period (T): 2π√(m/k) = 2π/ω
• Frequency (f): 1 / (2π) * √(k/m)

Where m is the mass of the system and k is the force constant.

Force (F): -kx

Acceleration (a): -ω²x

Maximum acceleration (|a|): ω²A

Equation of SHM: x(t) = A cos(ωt + φ)

Waves

Displacement (Y):

• Y = -A sin(kx - ωt + φ) (if wave is positive)
• Y = -A sin(kx + ωt + φ) (if wave is negative)

Wavelength (λ): 2π / k

Angular wave number (k): Propagation constant

Angular frequency (ω): 2π / T

Time period (T): 2π / ω

Law of Equipartition of Energy: For a system in thermal equilibrium, its total energy is divided equally among the degrees of freedom.