JEE Mains Physics · Formula sheet

Oscillations formulas

11 formulas, 1 reference table and 36 common traps for JEE Mains Physics Oscillations, grouped by subtopic.

SHM Equation, Velocity and Acceleration

Phase, velocity and acceleration from the SHM equation

Displacement, velocity and acceleration in SHM

x = A\sin(\omega t + \phi) \qquad v = A\omega\cos(\omega t + \phi) \qquad a = -\omega^{2}x

Speed at a displacement and amplitude from one snapshot

Speed and acceleration at a displacement

v = \omega\sqrt{A^{2} - x^{2}} \qquad v_{max} = A\omega \qquad a_{max} = \omega^{2}A

Common traps

At rest means phase π/2, not phase 0

At phase 0 the particle is at the mean position, moving at its fastest. It stops at the extremes, where the phase is π/2 or 3π/2.

The starting direction picks the initial phase

x(0) = A/2 gives sin φ = 1/2, which allows both π/6 and 5π/6. Moving towards +x needs cos φ > 0 (π/6); moving towards −x needs cos φ < 0 (5π/6).

Projection on the x-axis is a cosine

For a point on the reference circle at angle ωt + φ₀ from the x-axis, the projection on the x-axis is r cos(ωt + φ₀) and on the y-axis r sin(ωt + φ₀). Mixing them up shifts the phase by π/2.

Divide by the number in front of v² first

In 4v² = 50 − x², ω² is not 1. Divide through: v² = 12.5 − x²/4, so ω = 1/2 and the period is 4π, not 2π.

The v–x graph is an ellipse

Speed against displacement is an ellipse, not a straight line and not a parabola. It is a circle only when Aω equals A in the chosen units. The straight line is the a–x graph.

Acceleration is largest at the extremes

Where the speed is zero, the acceleration is ω²A, its largest value. At the mean position the speed is largest and the acceleration is zero.

Timing in SHM and Combining SHMs

Learn this subtopic in the notes

Time taken to move between two positions in SHM

Time from the phase covered

t = \frac{\Delta\theta}{\omega} = \frac{\Delta\theta}{2\pi}\,T

Adding two SHMs of the same frequency, and telling SHM from periodic motion

Resultant of two SHMs of the same frequency

A = \sqrt{A_1^{2} + A_2^{2} + 2A_1A_2\cos\Delta\phi} \qquad a\sin\omega t + b\cos\omega t = \sqrt{a^{2} + b^{2}}\,\sin\left(\omega t + \tan^{-1}\frac{b}{a}\right)

Common traps

Starting at an extreme needs the cosine

From x = A to x = A/2 is a phase of π/3 in x = A cos ωt, so it takes T/6. Reading it with x = A sin ωt gives T/12, which is the time from the mean to A/2, a different journey.

Equal distances do not take equal times

Mean to A/2 takes T/12, but A/2 to A takes T/6, twice as long, because the particle slows down near the extreme.

A quarter period covers A only from the mean or an extreme

Starting anywhere else, the distance in T/4 is not A. In a half period the distance is always 2A, wherever the motion starts.

Amplitudes add as vectors, not as numbers

Amplitudes 3 and 4 at a phase difference of π/2 give 5, not 7. Only SHMs exactly in phase give A₁ + A₂.

A constant shift does not spoil SHM

sin²ωt equals 1/2 − (1/2)cos 2ωt. That is SHM about x = 1/2, with angular frequency 2ω and period π/ω, half the period of sin ωt.

Two different frequencies never make SHM

cos ωt + cos 2ωt repeats every 2π/ω but is not SHM. If the two periods have an irrational ratio, as in sin ωt + cos πωt, the motion does not repeat at all.

Spring Systems and Restoring Forces

Learn this subtopic in the notes

Cut springs and springs in series and in parallel

Cut and combined springs

k \propto \frac{1}{l} \qquad \frac{1}{k_s} = \frac{1}{k_1} + \frac{1}{k_2} \qquad k_p = k_1 + k_2

Mass on a spring, and two masses on one spring

Spring–mass period and reduced mass

T = 2\pi\sqrt{\frac{m}{k}} \qquad \omega = \sqrt{\frac{k}{\mu}},\ \ \mu = \frac{m_1m_2}{m_1 + m_2}

SHM from any restoring force or torque

Angular frequency from a restoring force or torque

F = -Cx \Rightarrow \omega = \sqrt{\frac{C}{m}} \qquad \tau = -\kappa\theta \Rightarrow \omega = \sqrt{\frac{\kappa}{I}}

Common traps

A block between two walls is parallel

The springs sit on opposite sides of the block, so they look like a chain. But they stretch and compress by the same x and both push the block back: k = k₁ + k₂, not the series value.

Cutting a spring makes it stiffer

A shorter piece of the same spring has a larger constant. Cut in half, each half is 2k, not k/2.

Gravity along an incline does not change ω

For a block between springs on a smooth incline, mg sin α is a constant force. It shifts the equilibrium point but leaves the restoring force −(k₁ + k₂)x unchanged.

Two free masses use the reduced mass

With no wall, a spring between m₁ and m₂ oscillates with ω = √(k/μ), μ = m₁m₂/(m₁ + m₂). Using one mass or the total mass gives the wrong ω.

Gravity does not change a spring's period

Hanging the spring vertically shifts the equilibrium down by mg/k, but T = 2π√(m/k) is unchanged.

Equal amplitudes and equal top speeds give opposite ratios

For equal masses, v_max = A√(k/m). Equal amplitudes give v_max in the ratio √(k₁/k₂); equal v_max give amplitudes in the ratio √(k₂/k₁).

A physical pendulum uses I about the pivot

In T = 2π√(I/mgd), I is the moment of inertia about the pivot, found with the parallel-axis theorem. I about the centre of mass gives too short a period.

The tunnel period does not depend on the chord

The force along any chord through a uniform earth is −(mg/R)x, so every such tunnel gives T = 2π√(R/g), about 84 minutes.

A floating body's period depends on its depth under the liquid

T = 2π√(M/ρAg) equals 2π√(h/g), where h is the depth below the surface at rest. A denser liquid floats the block higher, so h is smaller and T is shorter.

Simple Pendulum and Effective g

Learn this subtopic in the notes

Period of a simple pendulum and how it depends on length

Simple pendulum

T = 2\pi\sqrt{\frac{L}{g}} \qquad \frac{\Delta T}{T} = \frac{1}{2}\frac{\Delta L}{L}

Effective g for a pendulum at a height, on a planet, in a lift or in a liquid

Pendulum with an effective g

T = 2\pi\sqrt{\frac{L}{g_{eff}}} \qquad g_h = g\left(\frac{R}{R + h}\right)^{2}

Common traps

A seconds pendulum has a period of 2 s

It takes 1 s to go from one extreme to the other, and 2 s for a full oscillation. Taking T = 1 s gives a length a quarter of the right one.

The bob's mass does not enter

Changing the mass of the bob, or keeping it the same, does not change T = 2π√(L/g). Only L and g matter.

A longer pendulum makes a clock slow

A longer pendulum has a longer period, so the clock ticks less often and loses time. Heat lengthens the pendulum, so a clock loses time in summer.

Large swings: the two accelerations point different ways

At the extreme the speed is zero, so the acceleration is only tangential, g sin θ₀. At the lowest point the tangential part is zero and only v²/L remains.

A height R puts the pendulum 2R from the centre

"At a height equal to the earth's radius" means a distance R + R = 2R from the centre, so g is g/4 and T doubles. Using distance R from the centre gives no change at all.

The lift's acceleration decides, not its velocity

A lift moving down but slowing has an upward acceleration, so g_eff = g + a and the period is shorter. Look at the direction of a, not of v.

A clock on a mountain runs slow

g is smaller at a height, so T is longer and the clock ticks less often. It runs slow, not fast.

Energy in SHM, Amplitude Changes and Damping

Learn this subtopic in the notes

Kinetic and potential energy at a displacement in SHM

Energy in SHM

E = \frac{1}{2}kA^{2} = \frac{1}{2}m\omega^{2}A^{2} \qquad U = \frac{1}{2}kx^{2} \qquad K = \frac{1}{2}k\left(A^{2} - x^{2}\right)

New amplitude after a push or an added mass, and decay by damping

Amplitude after a sudden change, and damped amplitude

A'^{2} = x^{2} + \frac{v'^{2}}{\omega'^{2}} \qquad A = A_0e^{-bt/2m}

Graphs and averages of energy in SHM

Quantity	Against displacement x	Against time t (starting at the mean)	Average over a period
Potential energy U	Upward parabola $\tfrac{1}{2}kx^{2}$ : zero at x = 0, E at $x = \pm A$	$\tfrac{E}{2}(1 - \cos 2\omega t)$ : zero at t = 0, peak E at T/4, repeats every T/2	$E/2$
Kinetic energy K	Downward parabola $\tfrac{1}{2}k(A^{2} - x^{2})$ : E at x = 0, zero at $x = \pm A$	$\tfrac{E}{2}(1 + \cos 2\omega t)$ : E at t = 0, zero at T/4, repeats every T/2	$E/2$ E − U against x is this same downward parabola.
Total energy E	Horizontal line at $\tfrac{1}{2}kA^{2}$	Horizontal line at $\tfrac{1}{2}kA^{2}$	$E$
Velocity v	Ellipse $\dfrac{x^{2}}{A^{2}} + \dfrac{v^{2}}{A^{2}\omega^{2}} = 1$	$A\omega\cos\omega t$ , repeats every T	Zero (average speed $2A\omega/\pi$ )
Acceleration a	Straight line $a = -\omega^{2}x$ through the origin	$-\omega^{2}A\sin\omega t$ , repeats every T	Zero

Energies repeat at twice the oscillator's frequency and never go below zero.

Common traps

The energies are equal at A/√2, not at A/2

At half the amplitude the potential energy is only a quarter of the total. Kinetic and potential are equal at x = A/√2, about 0.71A.

A spring's energy does not depend on the mass

E = ½kA² holds for any mass. A heavier block at the same amplitude has the same energy, a lower ω and a lower top speed.

"Energy of the block" at a point is the total

When a question gives the block's energy at some x, it means K + U, which equals ½kA². Do not set it equal to ½kx².

Energy oscillates at twice the frequency

If the kinetic energy varies at angular frequency Ω, the oscillator's own angular frequency is Ω/2. Taking them equal doubles the answer.

Potential energy is never negative

U = ½kx² is zero at the mean position and positive on both sides. A U–t graph that dips below zero, or a U–x graph that is a straight line, is wrong.

Where the mass is added decides the new amplitude

Added at the mean, momentum is shared and the amplitude drops to A√(M/(M + m)). Added at an extreme, the block is at rest there, so the amplitude stays A.

Multiply the speed, not the energy

Tripling the speed at x multiplies the kinetic energy there by 9. The new amplitude comes from A'² = x² + v'²/ω², not from tripling A.

Energy decays twice as fast as amplitude

A = A₀e^(−bt/2m) but E = E₀e^(−bt/m). The energy halves in half the time the amplitude takes.

Oscillations formulas

SHM Equation, Velocity and Acceleration

Phase, velocity and acceleration from the SHM equation

Speed at a displacement and amplitude from one snapshot

Common traps

Timing in SHM and Combining SHMs

Time taken to move between two positions in SHM

Adding two SHMs of the same frequency, and telling SHM from periodic motion

Common traps

Spring Systems and Restoring Forces

Cut springs and springs in series and in parallel

Mass on a spring, and two masses on one spring

SHM from any restoring force or torque

Common traps

Simple Pendulum and Effective g

Period of a simple pendulum and how it depends on length

Effective g for a pendulum at a height, on a planet, in a lift or in a liquid

Common traps

Energy in SHM, Amplitude Changes and Damping

Kinetic and potential energy at a displacement in SHM

New amplitude after a push or an added mass, and decay by damping

Graphs and averages of energy in SHM

Common traps

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