Simple Harmonic Motion (SHM) – Definition, Formulas & Energy | Entrance Exam Physics Notes

SIMPLE HARMONIC MOTION (SHM) — SHORT NOTES

1. Definition

A body performs Simple Harmonic Motion if:

• Acceleration is directly proportional to displacement, and

• Acceleration is always directed towards the mean position (opposite sign)

$$a\propto -x\Rightarrow a=-{\omega }^2\mathrm{x<br>}$$

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2. General Equation of SHM

Displacement

$$x=A\sin (\omega t+\varphi )$$

Where:

• $A$ = amplitude

• $\omega$ = angular frequency

• $\phi$ = phase constant

Velocity

$$v=\omega \sqrt{A^2-x^2}$$

Maximum velocity:

$$v_{max}=\omega A$$

Acceleration

$$a=-{\omega }^{2}x$$

Maximum acceleration:

$$a_{max}={\omega }^{2}A$$

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3. Time Period & Frequency

$$T=\frac{2\pi }{\omega },f=\frac{1}{T}$$

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4. Energy in SHM

Total Energy (constant)

$$E=\frac{1}{2}k{A}^2$$

Kinetic Energy

$$K=\frac{1}{2}m{\omega }^2(A^2-x^2)$$

Potential Energy

$$U=\frac{1}{2}m{\omega }^2{x}^2$$

At:

• Mean position → KE max, PE min

• Extreme position → KE zero, PE max

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5. Important SHM Systems

(a) Mass–Spring System

$$T=2\pi \sqrt{\frac{m}{k}}$$

(b) Simple Pendulum (small oscillations)

$$T=2\pi \sqrt{\frac{L}{g}}$$

Time period depends on length and gravity, not on mass or amplitude (for small angles).

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6. Phase & Phase Difference

• Phase describes the state of SHM at any time.

• Phase difference between two SHM motions:

  $$\mathrm{\Delta }\varphi ={\varphi }_2-{\varphi }_1$$

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7. Graphs (Important for Exams)

• Displacement vs Time → sine curve

• Velocity vs Time → cosine curve

• Acceleration vs Time → sine curve (opposite phase)

• Velocity is maximum at mean position

• Acceleration is maximum at extreme positions

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8. SHM as Projection of Uniform Circular Motion

A particle performing uniform circular motion → its projection on a diameter performs SHM.

Useful for deriving equations.

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9. Superposition of Two SHMs

Same frequency, different phasee:

$$A=\sqrt{A_1^2+A_2^2+2A_1{A}_2\cos \varphi }$$

Same amplitude, 90° phase difference:

$$A=\sqrt{2}A$$

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10. Damped & Forced Oscillations (Basic Points)

• Damped SHM: amplitude decreases due to friction/resistance

• Forced SHM: external periodic force applied

• Resonance: max amplitude when driving frequency = natural frequency

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11. Common Entrance Exam Problems

• SHM of mass‐spring systems

• Time period of pendulum & equivalent pendulum

• Energy distribution questions

• Maximum velocity/acceleration problems

• Phase difference questions

• Projection of UCM

• Superposition of SHMs