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Mass-Spring System
A mass-spring system consists of a mass attached to an elastic spring which can perform SHM when the spring obeys Hooke's law and damping is negligible.
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Mass-Spring System
A stiff spring pulls the mass back more strongly, so it oscillates faster.
A heavier mass is harder to accelerate, so it oscillates more slowly.
Recommended for Class 12
Mass-Spring System
Definition
A mass-spring system consists of a mass attached to an elastic spring which can perform SHM when the spring obeys Hooke's law and damping is negligible.
Dependence
Hooke's law:
**F = −kx**
Using Newton's second law:
**F = ma**
Therefore:
**ma = −kx**
or:
**a = −(km)x**
Compare with:
**a = −ω²x**
Thus:
**ω² = km**
and:
**ω = √(km)**
Therefore:
**T = 2π√(m/k)**
and:
**f = (1/2π)√(km)**
Increasing mass increases period.
Increasing spring constant decreases period.
Amplitude does not affect period for an ideal linear spring.
Formula / Key Relation
**F=-kx**
**ω=√(km)**
**T=2π√(m/k)**
**f=(1/2π)√(km)**
Mass-Spring System — Advanced
The equation of motion is:
**m(d²x/dt²)+kx=0**
which has angular frequency:
**ω=√(km)**
Common Mistakes
Common Mistakes
- Mistake
Do not write `T=2π√(km)`.
- Mistake
Correct:
- Mistake
`T=2π√(m/k)`
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Spring SHM follows directly from Hooke's law.
For an ideal spring, Hooke's law gives `F=-kx`. Combining it with `F=ma` gives `a=-(km)x`, which has the SHM form. Therefore `ω=√(km)` and `T=2π√(m/k)`. A heavier mass increases the period, while a stiffer spring decreases it.
- Definition
- For an ideal spring, Hooke's law gives `F=-kx`. Combining it with `F=ma` gives `a=-(km)x`, which has the SHM form. Therefore `ω=√(km)` and `T=2π√(m/k)`. A heavier mass increases the period, while a stiffer spring decreases it.
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Practice
PQ
Q1. State Hooke's law
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Q2. Derive angular frequency
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Q3. Write period
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Q4. What happens if mass quadruples?
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Q5. What happens if k quadruples?
Exam Practice
EQ
Q1. Derive the period of a mass-spring oscillator
EQ
Q2. Explain why spring motion can be SHM
EQ
Q3. Discuss dependence of T on mass
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Q4. Discuss dependence of T on spring constant
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Q5. Solve a numerical involving m, k, T or f
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.