Simple Pendulum
Deriving the time period of a simple pendulum for small oscillations and understanding factors affecting it.
What is Simple Pendulum?
An idealized system consisting of a point mass suspended by a massless, inextensible string from a rigid support, capable of oscillating freely.
Key formula / rule: Time Period of Simple Pendulum
Key points
- Define a simple pendulum and its components.
- Derive the expression for the time period of a simple pendulum for small oscillations.
- Identify the factors affecting the time period of a simple pendulum.
- Explain why the mass and amplitude do not affect the time period (for small amplitudes).
Common exam trap
Assuming the formula T = 2π√(l/g) is valid for large amplitudes.
Definitions
- Term
Simple Pendulum
- Meaning
An idealized system consisting of a point mass suspended by a massless, inextensible string from a rigid support, capable of oscillating freely.
- Term
Time Period (T)
- Meaning
The time taken by the pendulum to complete one full oscillation (back and forth motion).
- Term
Amplitude (A)
- Meaning
The maximum displacement of the pendulum bob from its equilibrium position.
Learning objectives
Define a simple pendulum and its components.
Derive the expression for the time period of a simple pendulum for small oscillations.
Identify the factors affecting the time period of a simple pendulum.
Explain why the mass and amplitude do not affect the time period (for small amplitudes).
Formulae
- Name
Time Period of Simple Pendulum
- Note
Valid for small angular displacements (θ < 15°).
- Expression
T = 2π√(l/g)
- Name
Angular Frequency of Simple Pendulum
- Note
Derived from ω = √(k/m) where k = mg/l.
- Expression
ω = √(g/l)
- Name
Restoring Force (approximate)
- Note
For small angles, where x is the tangential displacement.
- Expression
F ≈ -mgθ ≈ -(mg/l)x
Prerequisites
Understanding of Uniform Circular Motion.
Concept of Restoring Force.
Definition and characteristics of Simple Harmonic Motion (SHM).
Basic trigonometry (small angle approximation: sinθ ≈ θ).
Understanding of acceleration due to gravity (g).
Common mistakes
Assuming the formula T = 2π√(l/g) is valid for large amplitudes.
Confusing angular frequency (ω) with frequency (f).
Incorrectly assuming the time period depends on the mass of the bob.
Forgetting that 'l' is the length of the string, not the total length from support to the center of mass.
Keywords
Simple Pendulum
Oscillation
Simple Harmonic Motion
Time Period
Length
Acceleration due to Gravity
Amplitude
Restoring Force
Practice preview
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