Oscillations
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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SHM: Core Kinematics
Differential Equation
d²x/dt² + ω²x = 0
Condition: a ∝ -x
Condition: a ∝ -x
The fundamental condition for any motion to be Simple Harmonic.
Displacement (x)
x(t) = A sin(ωt + φ)
A: Amplitude
φ: Initial Phase
A: Amplitude
φ: Initial Phase
Describes the position of the particle at any time t.
Velocity (v)
v = dx/dt = Aω cos(ωt + φ)
v = ω√(A² - x²)
v = ω√(A² - x²)
Maximum at mean position (x=0), zero at extremes (x=±A).
Acceleration (a)
a = dv/dt = -Aω² sin(ωt + φ)
a = -ω²x
a = -ω²x
Maximum at extremes (x=±A), zero at mean position (x=0).
Angular Frequency (ω)
ω = √(k/m)
ω = 2πf = 2π/T
ω = 2πf = 2π/T
k is the spring constant or force constant of the system.
Time Period (T) & Frequency (f)
T = 2π/ω = 2π√(m/k)
f = 1/T
f = 1/T
Time period is independent of the amplitude for SHM.
Energy in SHM
Potential Energy (U)
U = ½kx²
U = ½mω²x²
U = ½mω²x²
Max at extremes, zero at mean position. Parabolic curve.
Kinetic Energy (K)
K = ½mv²
K = ½mω²(A² - x²)
K = ½mω²(A² - x²)
Max at mean position, zero at extremes. Inverted parabola.
Total Energy (E)
E = K + U
E = ½kA² = ½mω²A²
E = ½kA² = ½mω²A²
Total mechanical energy in SHM is always conserved.
Spring-Mass Systems
Horizontal & Vertical Spring
T = 2π√(m/k)
Time period is the same for both. In vertical case, equilibrium position shifts by mg/k.
Series Combination
1/kₛ = 1/k₁ + 1/k₂ + ...
T = 2π√(m/kₛ)
T = 2π√(m/kₛ)
Effective spring constant decreases, so time period increases.
Parallel Combination
kₚ = k₁ + k₂ + ...
T = 2π√(m/kₚ)
T = 2π√(m/kₚ)
Effective spring constant increases, so time period decreases.
BITSAT Speed Hacks
Recognize SHM: Any force F = -kx or potential U = ½kx² implies SHM. The coefficient of x in the acceleration equation `a = -ω²x` directly gives you ω².
Energy Shortcut: At position x = A/√2, Kinetic Energy = Potential Energy. This is a very common question.
Phase Logic: Velocity leads displacement by π/2. Acceleration leads velocity by π/2. Acceleration and displacement are out of phase (π).
Pendulum in Lift: For a lift accelerating up with 'a', use g_eff = g+a. For down, use g_eff = g-a. In free fall, T is infinite.
Resonance Condition: Max amplitude in forced oscillations occurs when driving frequency matches the natural frequency. Look for ω_drive = ω_natural.
Pendulums
Simple Pendulum
T = 2π√(L/g)
Valid for small angular displacements (θ < 10°). Independent of mass and amplitude.
Physical (Compound) Pendulum
T = 2π√(I / mgd)
I = Moment of inertia about pivot, d = distance from pivot to center of mass.
Seconds Pendulum
T = 2 seconds
L ≈ 99.4 cm (≈ 1m) on Earth
L ≈ 99.4 cm (≈ 1m) on Earth
A simple pendulum whose period is exactly two seconds (one second for a swing in one direction).
Damped & Forced Oscillations
Damped Oscillation
m(d²x/dt²) + b(dx/dt) + kx = 0
Amplitude: A(t) = A₀e-bt/2m
Amplitude: A(t) = A₀e-bt/2m
Amplitude decreases exponentially with time due to a resistive force (damping).
Angular Frequency (Damped)
ω' = √(k/m - (b/2m)²)
ω' = √(ω₀² - (b/2m)²)
ω' = √(ω₀² - (b/2m)²)
The frequency of damped oscillation is slightly less than the natural frequency ω₀.
Forced Oscillation & Resonance
m(d²x/dt²) + b(dx/dt) + kx = F₀cos(ωt)
Resonance occurs when driving frequency ω ≈ natural frequency ω₀, leading to maximum amplitude.
Quality Factor (Q)
Q = 2π (Energy Stored / Energy lost per cycle)
Q ≈ ω₀ / (b/m)
Q ≈ ω₀ / (b/m)
A high Q-factor implies low damping and a sharp, narrow resonance peak.
Oscillation Types at a Glance
| Parameter | Free (Undamped) SHM | Damped Oscillation | Forced Oscillation |
|---|---|---|---|
Amplitude | Constant (A) | Decreases exponentially (A₀e⁻ɣᵗ) | Constant in steady state |
Frequency | Natural frequency (ω₀) | Slightly less than ω₀ (ω') | Driving frequency (ω) |
Total Energy | Conserved (Constant) | Decreases exponentially | Constant (Input Rate = Loss Rate) |
External Force | Zero | Zero (only damping force) | Periodic Driving Force |