Time Period Formula & Stepwise Derivation for a Simple Pendulum
The time period of simple pendulum is a foundational physics concept relevant for JEE Main. It describes how long a pendulum bob takes to complete one full oscillation under gravity. Understanding its formula, dependencies, and derivation is crucial for acing oscillations and SHM-based questions in competitive exams.
Time Period of Simple Pendulum: Definition and Formula
The time period (t or T) is defined as the time taken by the bob to return to its original position after completing one full swing (oscillation). The standard formula is:
| Quantity | Symbol | SI Unit |
|---|---|---|
| Time Period | t or T | seconds (s) |
| Length of String | l | metre (m) |
| Acceleration due to Gravity | g | m/s2 |
Formula: t = 2π√(l/g) — where l is pendulum length, g is local gravity.
This means the time period depends only on the string length and gravitational acceleration, provided the oscillations are small. For dimensional analysis, [t] = [L1/2][L-1/2T] = [T].
Stepwise Derivation of the Time Period of Simple Pendulum
A simple pendulum performs simple harmonic motion (SHM) when displaced by a small angle θ from equilibrium. Restoring force F = –mgsinθ acts towards the mean position, with approximation sinθ ≈ θ (in radians) for small angles.
- Restoring torque: τ = –mglθ (for small θ)
- Equation of SHM: Iα = –mglθ
- Moment of inertia (I) for point mass: I = ml2
- α = d2θ/dt2
- So: ml2(d2θ/dt2) = –mglθ
- d2θ/dt2 + (g/l)θ = 0 (standard SHM form)
- Angular frequency ω = √(g/l)
- Time period: t = 2π/ω = 2π√(l/g)
The assumption of small-angle approximation is essential for accuracy in the formula. As per JEE Main, the pendulum's bob is treated as a point mass, and the string is assumed massless and inextensible.
Factors Affecting and Not Affecting Time Period of Simple Pendulum
Learners often ask which parameters influence the time period of simple pendulum and which do not. The clarity is crucial for exam success.
| Factor | Effect |
|---|---|
| Length of string (l) | Directly affects period (t ∝ √l) |
| Acceleration due to gravity (g) | Inversely affects period (t ∝ 1/√g) |
| Mass of bob | No effect (independent) |
| Amplitude (for small angles) | No effect (independent, if θ ≤ 5°) |
| Material/Shape of bob | No effect (as long as it's a point mass) |
Common misconception: The period does not depend on the mass or material of the bob, nor on the amplitude (as long as oscillations are small).
Simple Pendulum: Experimental Setup and Procedure
To measure the time period of simple pendulum practically, a point-mass bob is suspended by a light, non-stretchable string fixed at one end. The bob is displaced by a small angle and released to oscillate. For data accuracy, record the time for 20–30 oscillations and find average time per oscillation.
- Measure string length (l) from suspension point to bob center
- Use a stop-clock to record total time for 'n' oscillations
- Calculate time period: t = total time / n
- Repeat by changing l to plot t2 vs. l (linear graph)
- Typical errors: large amplitude swings, parallax, uncertainty counting oscillations
- Reduce friction and air resistance for accurate results
This experimental method directly tests the formula. To study measurement errors for JEE, review error analysis concepts from prior Vedantu resources.
Solved Examples: Time Period of Simple Pendulum Calculations
Example 1: Calculate the time period for a simple pendulum of length 1.0 m, with g = 9.8 m/s2.
- t = 2π√(l/g) = 2π√(1.0/9.8)
- = 2π × 0.319
- = 2.006 seconds
- Final answer: 2.01 s (rounded to two decimal places)
Example 2: If the length of a pendulum is made four times, what happens to the time period?
- t ∝ √l → New t = 2 × Original t
- The period doubles if length is quadrupled.
Solved numericals clarify how to use the formula and help avoid calculation errors. For SHM comparison, see time period of SHM for springs and other systems.
Real-Life Applications and Importance
Understanding the time period of simple pendulum helps in:
- Designing pendulum clocks and precise timekeeping devices
- Laboratory measurement of gravity g
- Physics experiments verifying SHM and oscillations concepts
- Demonstrating periodic motion in classroom setups
The seconds pendulum (T = 2s) was historically vital for defining the standard second. Knowledge of this principle is foundational for understanding periodic motion, vital for broader mechanics topics covered on Vedantu.
Summary Table: Key Points on Time Period of Simple Pendulum
| Point | Quick Recap |
|---|---|
| Governing formula | t = 2π√(l/g) |
| Dimension | [T] |
| Depends on | Length l, gravity g |
| Independent of | Mass and (for small angles) amplitude |
| Experiment errors | Parallax, counting error, large amplitude |
| Pendulum type in clocks | Seconds pendulum (T = 2 s) |
For further reading, explore linked resources on simple pendulum theory, apply oscillation principles in depth, and connect to SHM derivations. Refer to Vedantu's detailed guides for JEE revision on these topics.
FAQs on Time Period of a Simple Pendulum Explained
1. What is the formula for the time period of a simple pendulum?
The time period (T) of a simple pendulum is given by the formula: T = 2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity.
Key points:
- This formula applies for small angular displacements (small amplitude).
- The time period does not depend on the mass or material of the bob.
- It is derived assuming the pendulum behaves as a simple harmonic oscillator.
2. What does the time period of a simple pendulum depend on?
The time period of a simple pendulum depends only on its length and the acceleration due to gravity.
- Directly proportional to the square root of the length (l).
- Inversely proportional to the square root of g (acceleration due to gravity).
- Independent of the mass and material of the bob (for small angles).
- Approximately independent of amplitude, provided the angle is small (<15°).
3. Does the mass of the bob affect the time period of a simple pendulum?
No, the mass of the bob does not affect the time period of a simple pendulum.
- The formula T = 2π√(l/g) shows no dependence on mass.
- This feature demonstrates that all pendulums of equal length oscillate at the same rate, regardless of bob weight, if amplitude is small.
4. What is meant by the time period of a pendulum?
The time period of a pendulum is the time taken to complete one full oscillation (back and forth) around its rest position.
- The concept of time period is crucial in classifying motion as periodic or oscillatory.
- It is often measured in seconds using a stopwatch during practical experiments.
5. How is the time period measured in a pendulum experiment?
The time period is measured by recording the time taken for a set number of oscillations and dividing by the number of oscillations.
- Set the pendulum to oscillate through a small angle.
- Count n complete oscillations (e.g., 20 swings).
- Measure the total time (t) using a stopwatch.
- Calculate Time period (T) = t/n.
6. What is a seconds pendulum?
A seconds pendulum is a simple pendulum whose time period is exactly 2 seconds.
- It completes one full oscillation in 2 seconds: 1 second for each swing (forward or backward).
- The length of a seconds pendulum on Earth (at standard gravity) is approximately 99.3 cm.
7. What is the dimensional formula of the time period of a simple pendulum?
The dimensional formula of the time period (T) of a simple pendulum is [T].
- From T = 2π√(l/g):
- l has dimensions [L], g has dimensions [L T-2].
- So, T = 2π√(L / [L T-2]) = 2π√(T2) = [T].
8. Can you give an example calculation for the time period of a simple pendulum?
Yes, here's a sample calculation:
- Let the length (l) = 1 m
- Acceleration due to gravity (g) = 9.8 m/s2
- Use the formula: T = 2π√(l/g)
- Calculate: √(1/9.8) ≈ 0.319
- T = 2 × 3.14 × 0.319 ≈ 2.006 s
9. Why is the time period of a simple pendulum independent of its amplitude?
For small amplitudes, the time period of a simple pendulum is independent of amplitude due to the mathematics of simple harmonic motion.
- This approximation uses sin θ ≈ θ for small angles (<15°).
- At larger amplitudes, the period increases slightly because the motion is no longer perfectly simple harmonic.
10. What happens to the time period if the length of the pendulum is doubled?
If the length of the pendulum is doubled, the time period increases by a factor of √2.
- Original period: T₁ = 2π√(l/g)
- New length: 2l
- New period: T₂ = 2π√(2l/g) = T₁ × √2
11. What are common errors in the simple pendulum experiment?
Common errors in the simple pendulum experiment include:
- Large angle swings (amplitude too big)
- Inaccurate length measurement (not measuring from suspension point to center of bob)
- Air resistance and friction effects
- Human error in stopwatch operation
- Counting wrong number of oscillations
12. Is the simple pendulum an example of periodic or simple harmonic motion?
The simple pendulum is an example of both periodic and simple harmonic motion (SHM) for small angles.
- Its motion repeats itself at regular intervals (periodic).
- For small displacements, the restoring force is proportional to displacement, satisfying the conditions for SHM.
