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SHM formulas

Master SHM through 32 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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SHM, every formula

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SHM restoring force
$F = - k x$
Q1MCQRestoring force
The defining condition for SHM is:
- A
  $F=-kx$
- B
  $F=kx$
- C
  $F=-kx^{2}$
- D
  $F=k$
General equation of SHM
$x = A sin (ω t + ϕ)$
Q1MCQGeneral equation
The displacement in SHM is:
- A
  $A\sin(\omega t+\phi)$
- B
  $A e^{\omega t}$
- C
  $A\omega t$
- D
  $A t^{2}$
Angular frequency
$ω = \frac{2 π}{T} = 2 π f$
Velocity in SHM
$v = ω A^{2} - x^{2}$
Q1MCQVelocity
The speed of a particle in SHM at displacement $x$ is:
- A
  $\omega\sqrt{A^{2}-x^{2}}$
- B
  $\omega A$
- C
  $\omega x$
- D
  $\omega^{2}x$
Maximum velocity
$v_{m a x} = ω A$
Acceleration in SHM
$a = - ω^{2} x$
Q1MCQAcceleration
The acceleration in SHM is:
- A
  $-\omega^{2}x$
- B
  $-\omega x$
- C
  $\omega^{2}A$
- D
  $-kx^{2}$
Maximum acceleration
$a_{m a x} = ω^{2} A$
Kinetic energy
$K = \frac{1}{2} m ω^{2} (A^{2} - x^{2})$
Q1MCQKinetic energy
The KE of a particle in SHM is:
- A
  $\tfrac12 m\omega^{2}(A^{2}-x^{2})$
- B
  $\tfrac12 m\omega^{2}x^{2}$
- C
  $\tfrac12 kA^{2}$
- D
  $0$
Potential energy
$U = \frac{1}{2} k x^{2} = \frac{1}{2} m ω^{2} x^{2}$
Q1MCQPotential energy
The PE of a particle in SHM is:
- A
  $\tfrac12 kx^{2}$
- B
  $\tfrac12 kA^{2}$
- C
  $\tfrac12 m\omega^{2}(A^{2}-x^{2})$
- D
  $0$
Total energy
$E = \frac{1}{2} m ω^{2} A^{2} = \frac{1}{2} k A^{2}$
Q1MCQTotal energy
The total energy in SHM is:
- A
  $\tfrac12 kA^{2}$
- B
  $\tfrac12 kx^{2}$
- C
  depends on $x$
- D
  $0$
Spring–mass period
$T = 2 π \frac{m}{k}$
Q1MCQSpring period
The period of a spring–mass system is:
- A
  $2\pi\sqrt{\dfrac{m}{k}}$
- B
  $2\pi\sqrt{\dfrac{k}{m}}$
- C
  $2\pi\dfrac{m}{k}$
- D
  $2\pi\sqrt{mk}$
Angular frequency (spring)
$ω = \frac{k}{m}$
Two-mass spring period
$T = 2 π \frac{μ}{k}$
μ = reduced mass
Q1MCQTwo-mass spring
For two masses joined by a spring, the period uses:
- A
  the reduced mass $\mu$
- B
  the total mass
- C
  the larger mass
- D
  $m_1-m_2$
Springs in series
$\frac{1}{k _{e q}} = \frac{1}{k _{1}} + \frac{1}{k _{2}}$
Springs in parallel
$k_{e q} = k_{1} + k_{2}$
Simple pendulum period
$T = 2 π \frac{l}{g}$
Q1MCQSimple pendulum
The period of a simple pendulum is:
- A
  $2\pi\sqrt{\dfrac{l}{g}}$
- B
  $2\pi\sqrt{\dfrac{g}{l}}$
- C
  $2\pi\dfrac{l}{g}$
- D
  $2\pi\sqrt{lg}$
Second's pendulum
$T = 2 s, l \approx 1 m$
Physical (compound) pendulum
$T = 2 π \frac{I}{m g l}$
Q1MCQPhysical pendulum
The period of a compound (physical) pendulum is:
- A
  $2\pi\sqrt{\dfrac{I}{mgl}}$
- B
  $2\pi\sqrt{\dfrac{l}{g}}$
- C
  $2\pi\sqrt{\dfrac{I}{C}}$
- D
  $2\pi\sqrt{\dfrac{mgl}{I}}$
Torsional pendulum
$T = 2 π \frac{I}{C}$
C = torsional constant
Q1MCQTorsional pendulum
The period of a torsional pendulum is:
- A
  $2\pi\sqrt{\dfrac{I}{C}}$
- B
  $2\pi\sqrt{\dfrac{C}{I}}$
- C
  $2\pi\sqrt{\dfrac{I}{mgl}}$
- D
  $2\pi\sqrt{\dfrac{l}{g}}$
Conical pendulum period
$T = 2 π \frac{h}{g}$
Q1MCQConical pendulum
The period of a conical pendulum of vertical height $h$ is:
- A
  $2\pi\sqrt{\dfrac{h}{g}}$
- B
  $2\pi\sqrt{\dfrac{l}{g}}$
- C
  $2\pi\sqrt{\dfrac{g}{h}}$
- D
  $2\pi\sqrt{\dfrac{h}{2g}}$
Pendulum in a lift (up)
$T = 2 π \frac{l}{g + a}$
Q1MCQPendulum lift up
In a lift accelerating up at $a$, the pendulum period is:
- A
  $2\pi\sqrt{\dfrac{l}{g+a}}$
- B
  $2\pi\sqrt{\dfrac{l}{g-a}}$
- C
  $2\pi\sqrt{\dfrac{l}{g}}$
- D
  $2\pi\sqrt{\dfrac{l}{a}}$
Pendulum in a lift (down)
$T = 2 π \frac{l}{g - a}$
Q1MCQPendulum lift down
In a lift accelerating down at $a$, the pendulum period is:
- A
  $2\pi\sqrt{\dfrac{l}{g-a}}$
- B
  $2\pi\sqrt{\dfrac{l}{g+a}}$
- C
  $2\pi\sqrt{\dfrac{l}{g}}$
- D
  $2\pi\sqrt{\dfrac{l}{a}}$
Effective g (horizontal a)
$g_{e f f} = g^{2} + a^{2}$
Q1MCQEffective g
For a lift accelerating horizontally at $a$, the effective gravity is:
- A
  $\sqrt{g^{2}+a^{2}}$
- B
  $g+a$
- C
  $g-a$
- D
  $g$
Average KE over a cycle
$⟨ K ⟩ = \frac{1}{4} k A^{2}$
Q1MCQAverage KE
The time-average of KE over a cycle is:
- A
  $\tfrac14 kA^{2}$
- B
  $\tfrac12 kA^{2}$
- C
  $kA^{2}$
- D
  $0$
Average PE over a cycle
$⟨ U ⟩ = \frac{1}{4} k A^{2}$
Q1MCQAverage PE
The time-average of PE over a cycle is:
- A
  $\tfrac14 kA^{2}$
- B
  $\tfrac12 kA^{2}$
- C
  $kA^{2}$
- D
  $0$
Frequency of energy
$f_{e n er g y} = 2 f_{d i s pl a ce m e n t}$
Q1MCQEnergy frequency
The frequency of KE/PE variation compared with the displacement frequency is:
- A
  twice
- B
  half
- C
  equal
- D
  quadruple
Superposition amplitude
$A = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} cos θ$
Q1MCQSuperposition
The resultant amplitude of two SHMs of amplitudes $A_1,A_2$ with phase $\theta$ is:
- A
  $\sqrt{A_1^{2}+A_2^{2}+2A_1A_2\cos\theta}$
- B
  $A_1+A_2$
- C
  $\sqrt{A_1^{2}+A_2^{2}}$
- D
  $A_1A_2$
Damping force
$F = - b v$
Q1MCQDamping force
The damping force in a damped oscillator is:
- A
  $-b\vec v$
- B
  $-kx$
- C
  $-b\vec x$
- D
  $mg$
Damped amplitude
$A = A_{0} e^{- b t /2 m}$
Q1MCQDamped amplitude
The amplitude of a damped oscillator decays as:
- A
  $A_0 e^{-bt/2m}$
- B
  $A_0 e^{-bt/m}$
- C
  $A_0 e^{-t}$
- D
  $A_0$
Damped energy
$E (t) = \frac{1}{2} k A_{0}^{2} e^{- b t / m}$
Q1MCQDamped energy
The energy of a damped oscillator decays as:
- A
  $e^{-bt/m}$
- B
  $e^{-bt/2m}$
- C
  $e^{-t}$
- D
  constant
Critical damping
$b^{2} = 4 mk$
Q1MCQCritical damping
Critical damping occurs when:
- A
  $b^{2}=4mk$
- B
  $b^{2}>4mk$
- C
  $b^{2}<4mk$
- D
  $b=0$
Resonance amplitude
$A = \frac{F _{0}}{ω _{d} b}$
at resonance
Q1MCQResonance amplitude
At resonance, the driven amplitude is:
- A
  $\dfrac{F_0}{\omega_d b}$
- B
  $\dfrac{F_0}{m}$
- C
  $0$
- D
  $\dfrac{F_0}{k}$

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SHM formulas

SHM, every formula

SHM restoring force

General equation of SHM

Angular frequency

Velocity in SHM

Maximum velocity

Acceleration in SHM

Maximum acceleration

Kinetic energy

Potential energy

Total energy

Spring–mass period

Angular frequency (spring)

Two-mass spring period

Springs in series

Springs in parallel

Simple pendulum period

Second's pendulum

Physical (compound) pendulum

Torsional pendulum

Conical pendulum period

Pendulum in a lift (up)

Pendulum in a lift (down)

Effective g (horizontal a)

Average KE over a cycle

Average PE over a cycle

Frequency of energy

Superposition amplitude

Damping force

Damped amplitude

Damped energy

Critical damping

Resonance amplitude

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