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Simple Pendulum Period Lab Guide

PhysicsBeginnerReading time: 3 min

Overview

The simple pendulum is one of the most elegant periodic motion models in physics. This experiment uses the control variables method to explore the relationship between pendulum period and length, bob mass, and amplitude, verifying the period formula T=2πLgT = 2\pi\sqrt{\frac{L}{g}} and understanding that the period depends only on the pendulum length.

Background

The study of pendulums began with Galileo. In 1583, the 19-year-old Galileo observed a swinging chandelier in the Pisa Cathedral and timed it using his own pulse, discovering that regardless of the amplitude, each swing seemed to take the same time—this was the famous discovery of 'isochronism'. Later, Dutch physicist Christiaan Huygens invented the pendulum clock in 1656 using this principle, greatly improving timekeeping precision and ushering in a new era of accurate time measurement. The rigorous derivation of the pendulum period formula required the establishment of Newtonian mechanics.

Key Concepts

Simple Pendulum

An idealized model consisting of an inextensible, massless string with a small bob suspended from its end. The mass of the string and air resistance are neglected.

Period

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

The time required for the bob to complete one full back-and-forth oscillation, denoted by TT, measured in seconds (s).

Pendulum Length

The distance from the pivot point to the center of mass of the bob, denoted by LL, measured in meters (m).

Small Angle Approximation

sin⁡θ≈θ (when θ<15°)\sin\theta \approx \theta \text{ (when } \theta < 15° \text{)}

When the angle θ\theta is small (typically less than 15°15°), sin⁡θ≈θ\sin\theta \approx \theta (in radians), and the pendulum undergoes simple harmonic motion, making the period formula valid.

Formulas & Derivation

Simple Pendulum Period Formula

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

Relationship Between Period and Length

T∝LT \propto \sqrt{L}

Experiment Steps

  1. 1

    Investigate the Relationship Between Period and Length

    Keep mass (e.g., 100 g100\ \text{g}) and angle (e.g., 10°10°) constant. Set the length to 0.25 m0.25\ \text{m}, 0.50 m0.50\ \text{m}, and 1.00 m1.00\ \text{m} successively, release the pendulum, and record the measured period. Observe: When the length is quadrupled, how does the period change?
  2. 2

    Investigate the Relationship Between Period and Mass

    Keep length (e.g., 0.50 m0.50\ \text{m}) and angle (e.g., 10°10°) constant. Set the mass to 50 g50\ \text{g}, 200 g200\ \text{g}, and 500 g500\ \text{g} successively, release the pendulum, and record the measured period. Observe: Does the period change when the bob mass is varied?
  3. 3

    Investigate the Relationship Between Period and Amplitude

    Keep length (e.g., 0.50 m0.50\ \text{m}) and mass (e.g., 100 g100\ \text{g}) constant. Set the initial angle to 5°5°, 10°10°, and 15°15° successively, release the pendulum, and record the measured period. Observe: Within the small angle range, does the period change noticeably when amplitude is varied?
  4. 4

    Verify the Period Formula

    Choose a set of parameters (e.g., L=1.00 mL = 1.00\ \text{m}), calculate the theoretical period using T=2πLgT = 2\pi\sqrt{\frac{L}{g}}, and compare with the measured value. Do they match?

Learning Outcomes

  • Understand that the simple pendulum period depends only on the length and gravitational acceleration, not on the bob mass or amplitude
  • Master the application of the period formula T=2πLgT = 2\pi\sqrt{\frac{L}{g}}
  • Learn to use the control variables method to design experiments investigating each factor's effect on the period
  • Understand the physical model of simple harmonic motion under small angle approximation

Real-world Applications

  • Pendulum Clocks: Traditional pendulum clocks use the isochronism principle for precise timekeeping, adjusting the pendulum length to calibrate clock speed
  • Measuring Gravitational Acceleration: By measuring the pendulum period and length, the local gravitational acceleration can be calculated: g=4π2LT2g = \frac{4\pi^2 L}{T^2}
  • Seismometers: Early seismometers used long-period pendulums to detect small ground vibrations
  • Metronomes: Musical metronomes use adjustable-length pendulums to produce steady beats

Common Misconceptions

Misconception
A heavier bob results in a longer period
Correct
The pendulum period is independent of the bob mass. Although a heavier bob experiences greater gravitational force, its inertia is also greater, and these effects cancel out.
Misconception
A larger amplitude results in a longer period
Correct
Within the small angle range (<15°< 15°), the pendulum period is essentially independent of amplitude (isochronism). Only when the angle is very large does the period increase slightly.
Misconception
The pendulum length is the length of the string
Correct
The pendulum length is the distance from the pivot point to the center of mass of the bob, which includes the string length plus the bob radius (for a uniform sphere).

Further Reading

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