The Simple Pendulum
Derivation of the time period for a simple pendulum, understanding its assumptions and limitations for SHM.
What is The Simple Pendulum?
An idealized system consisting of a point mass (bob) suspended by a massless, inextensible string of length L from a rigid support, which oscillates under gravity.
Key formula / rule: Time Period of a Simple Pendulum
Key points
- Define a simple pendulum and its ideal conditions.
- Derive the expression for the time period of a simple pendulum.
- Identify the factors affecting the time period.
- Explain the conditions under which a simple pendulum exhibits SHM.
Common exam trap
Assuming SHM for large angular displacements.
Definitions
- Term
Simple Pendulum
- Meaning
An idealized system consisting of a point mass (bob) suspended by a massless, inextensible string of length L from a rigid support, which oscillates under gravity.
- Term
Time Period (T)
- Meaning
The time taken by the pendulum to complete one full oscillation (back and forth).
- Term
Amplitude
- Meaning
The maximum displacement of the pendulum bob from its equilibrium position.
- Term
Simple Harmonic Motion (SHM)
- Meaning
A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
Learning objectives
Define a simple pendulum and its ideal conditions.
Derive the expression for the time period of a simple pendulum.
Identify the factors affecting the time period.
Explain the conditions under which a simple pendulum exhibits SHM.
Differentiate between ideal and real pendulums.
Formulae
- Name
Time Period of a Simple Pendulum
- Note
Where T is the time period, L is the length of the pendulum, and g is the acceleration due to gravity. Valid for small angular displacements.
- Expression
T = 2π√(L/g)
- Name
Frequency of a Simple Pendulum
- Note
Where f is the frequency of oscillation.
- Expression
f = 1/T = (1/2π)√(g/L)
- Name
Angular Frequency of a Simple Pendulum
- Note
Where ω is the angular frequency.
- Expression
ω = √(g/L)
Prerequisites
Understanding of circular motion.
Concept of oscillations and periodic motion.
Basic trigonometry (small angle approximation).
Newton's laws of motion.
Concept of acceleration due to gravity.
Common mistakes
Assuming SHM for large angular displacements.
Confusing the formula for time period with frequency.
Incorrectly assuming the time period depends on the mass of the bob or amplitude.
Using incorrect units for L or g.
Forgetting the square root in the time period formula.
Keywords
Simple Pendulum
Oscillation
Periodic Motion
Simple Harmonic Motion (SHM)
Time Period
Frequency
Length
Gravity
Amplitude
Restoring Force
Practice preview
A simple pendulum has a time period T. If its length is made 1/4th, what will be its new time period?…
medium
A simple pendulum of length L has a time period T. If the mass of the bob is doubled, the new time period will be:…
easy
What is the time period of a simple pendulum of length L?…
easy