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Simple Pendulum
A simple pendulum consists ideally of a point mass suspended by a light inextensible string which performs approximately SHM for small angular displacements.
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Simple Pendulum
A pendulum swings because gravity keeps pulling it back toward its middle position.
For small swings, this restoring effect produces nearly SHM.
Recommended for Class 12
Simple Pendulum
Definition
A simple pendulum consists ideally of a point mass suspended by a light inextensible string which performs approximately SHM for small angular displacements.
Important Results
For pendulum length `L` and angular displacement `θ`, the restoring component of weight is:
**F = −mg sinθ**
For small angles measured in radians:
**sinθ ≈ θ**
Since:
**x ≈ Lθ**
the restoring effect becomes proportional to displacement.
Therefore the small-angle pendulum performs approximate SHM.
Angular frequency:
**ω = √(g/L)**
Period:
**T = 2π√(Lg)**
Frequency:
**f=(1/2π)√(g/L)**
Period increases with pendulum length.
Period decreases when g increases.
For small oscillations, period is approximately independent of mass.
Formula / Key Relation
**T=2π√(Lg)**
**ω=√(g/L)**
Simple Pendulum — Advanced
The exact pendulum equation is nonlinear:
**d²θ/dt² +(g/L)sinθ=0**
Using the small-angle approximation `sinθ≈θ` converts it into the SHM equation.
Common Mistakes
Common Mistakes
- Mistake
Do not include bob mass in the period formula.
- Mistake
Do not use the small-angle formula blindly for very large amplitudes.
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A small-angle simple pendulum approximately performs SHM.
A simple pendulum consists ideally of a point mass on a light string. Gravity provides its restoring force. The exact restoring force contains `sinθ`, but for small angles `sinθ≈θ`, making restoring acceleration proportional to displacement. The pendulum therefore approximates SHM with `ω=√(g/L)` and `T=2π√(Lg)`. Period depends on length and gravitational acceleration but not on bob mass.
- Definition
- A simple pendulum consists ideally of a point mass on a light string. Gravity provides its restoring force. The exact restoring force contains `sinθ`, but for small angles `sinθ≈θ`, making restoring acceleration proportional to displacement. The pendulum therefore approximates SHM with `ω=√(g/L)` and `T=2π√(Lg)`. Period depends on length and gravitational acceleration but not on bob mass.
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Practice
PQ
Q1. State pendulum period formula
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Q2. Why is small-angle approximation needed?
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Q3. Does bob mass affect ideal period?
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Q4. What happens if L quadruples?
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Q5. What happens if g quadruples?
Exam Practice
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Q1. Derive the period of a simple pendulum for small oscillations
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Q2. Explain the small-angle approximation
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Q3. Discuss dependence of period on L and g
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Q4. Explain why bob mass does not appear in the period
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Q5. Solve a pendulum numerical
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.