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Simple Harmonic Motion
Simple harmonic motion (SHM) is periodic motion in which the acceleration of an object is directly proportional to its displacement from the equilibrium position and is always directed toward equilibrium.
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Simple Harmonic Motion
Imagine pulling a spring slightly and releasing it.
The object moves toward the middle, passes it, reaches the other side and returns.
If the restoring effect increases directly with distance from the middle, the motion is SHM.
Recommended for Class 12
Simple Harmonic Motion
Definition
Simple harmonic motion (SHM) is periodic motion in which the acceleration of an object is directly proportional to its displacement from the equilibrium position and is always directed toward equilibrium.
At Equilibrium
An oscillating body moves repeatedly about an equilibrium or mean position.
Not every oscillation is SHM.
For motion to be simple harmonic:
a ∝ −x
Therefore:
**a = −ω²x**
where:
`a` = acceleration `x` = displacement from equilibrium `ω` = angular frequency
The negative sign shows that acceleration is always opposite to displacement.
`x = 0`
Therefore:
`a = 0`
Speed is maximum.
At Extreme Position
`x = ±A`
Acceleration magnitude is maximum.
Speed is zero.
Restoring Force
Using `F = ma`:
**F = −mω²x**
Thus restoring force is also proportional to displacement and directed toward equilibrium.
Formula / Key Relation
**a = −ω²x**
**F = −mω²x**
**x = A cos(ωt+φ)**
**ω = 2πf = 2π/T**
Simple Harmonic Motion — Advanced
SHM is the solution of the differential equation:
**d²x/dt² + ω²x = 0**
Its general displacement can be represented by:
**x = A cos(ωt + φ)**
or an equivalent sine form.
Common Mistakes
Common Mistakes
- Mistake
Do not define every periodic motion as SHM.
- Mistake
Do not forget the negative sign in:
- Mistake
`a = −ω²x`
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
SHM is an oscillation controlled by a restoring acceleration directed toward equilibrium.
Simple harmonic motion is periodic motion about an equilibrium position in which acceleration is directly proportional to displacement but opposite in direction. Its defining equation is `a = −ω²x`. At equilibrium, displacement and acceleration are zero while speed is maximum. At an extreme position, speed becomes zero while acceleration magnitude is maximum. SHM can be represented by `x=A cos(ωt+φ)`.
- Definition
- Simple harmonic motion is periodic motion about an equilibrium position in which acceleration is directly proportional to displacement but opposite in direction. Its defining equation is `a = −ω²x`. At equilibrium, displacement and acceleration are zero while speed is maximum. At an extreme position, speed becomes zero while acceleration magnitude is maximum. SHM can be represented by `x=A cos(ωt+φ)`.
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Practice
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Q1. Define SHM
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Q2. Write the defining equation
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Q3. What happens to velocity at equilibrium?
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Q4. What happens to acceleration at an extreme?
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Q5. What does the negative sign mean?
Exam Practice
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Q1. Define simple harmonic motion and state its necessary condition
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Q2. Explain restoring force in SHM
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Q3. Describe velocity and acceleration at equilibrium and extreme positions
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Q4. Derive `F=-mω²x`
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Q5. Explain why all periodic motions are not SHM
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.