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Displacement, Velocity and Acceleration in SHM
In SHM, displacement, velocity and acceleration vary periodically with time and have fixed mathematical relationships.
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Displacement, Velocity and Acceleration in SHM
At the middle:
speed is fastest, acceleration is zero.
At the ends:
speed is zero, acceleration is strongest.
Recommended for Class 12
Displacement, Velocity and Acceleration in SHM
Definition
In SHM, displacement, velocity and acceleration vary periodically with time and have fixed mathematical relationships.
Formula / Key Relation
Take:
**x=A cos(ωt+φ)**
Differentiate:
**v=dx/dt**
Therefore:
**v=−Aω sin(ωt+φ)**
Velocity in terms of displacement:
**v²=ω²(A²−x²)**
Hence:
**v=±ω√(A²−x²)**
Maximum velocity:
**vmax=Aω**
Differentiate velocity:
**a=dv/dt**
Therefore:
**a=−Aω²cos(ωt+φ)**
Since:
`x=Acos(ωt+φ)`
we obtain:
**a=−ω²x**
Maximum acceleration magnitude:
**amax=Aω²**
**x=Acos(ωt+φ)**
**v=−Aωsin(ωt+φ)**
**v²=ω²(A²−x²)**
**a=−ω²x**
**vmax=Aω**
**amax=Aω²**
Displacement, Velocity and Acceleration in SHM — Advanced
Velocity is a quarter-cycle out of phase with displacement, while acceleration is opposite in phase to displacement.
Common Mistakes
Common Mistakes
- Mistake
Do not say velocity is maximum at extreme positions.
- Mistake
Do not omit the negative sign from acceleration.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Velocity peaks at equilibrium; acceleration peaks at extremes.
For SHM, displacement can be written as `x=Acos(ωt+φ)`. Differentiation gives `v=-Aωsin(ωt+φ)` and then `a=-ω²x`. Maximum speed is `Aω` and maximum acceleration magnitude is `Aω²`. At equilibrium speed is maximum and acceleration zero; at extreme positions speed is zero and acceleration magnitude maximum.
- Definition
- For SHM, displacement can be written as `x=Acos(ωt+φ)`. Differentiation gives `v=-Aωsin(ωt+φ)` and then `a=-ω²x`. Maximum speed is `Aω` and maximum acceleration magnitude is `Aω²`. At equilibrium speed is maximum and acceleration zero; at extreme positions speed is zero and acceleration magnitude maximum.
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Practice
PQ
Q1. Write displacement equation
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Q2. Write velocity equation
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Q3. Write acceleration equation
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Q4. Where is speed maximum?
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Q5. Where is acceleration magnitude maximum?
Exam Practice
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Q1. Derive velocity from the SHM displacement equation
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Q2. Derive acceleration and prove `a=-ω²x`
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Q3. Derive velocity-displacement relation
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Q4. Find maximum velocity and acceleration
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Q5. Compare x, v and a at equilibrium and extremes
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.