Pendulum & Spring SHM - A-level Physics

Simple Harmonic Motion: Ball on a String and Mass on a Spring

Key Concepts: Simple Harmonic Motion (SHM), Pendulum (Ball on a String), Mass on a Spring, Time Period (T), Length (L), Gravitational Field Strength (G), Mass (M), Spring Constant (K), Proportionality, Experimental Determination of G and K, Fiducial Marker, Oscillations.

Introduction to SHM Examples

The video focuses on two specific examples of Simple Harmonic Motion (SHM): a ball on a string (pendulum) and a mass on a spring. These examples are fundamental to understanding many physics concepts.

Time Period Equations

• Pendulum (Ball on a String): T = 2π * √(L/G), where:
  • T is the time period.
  • L is the length of the string.
  • G is the gravitational field strength (acceleration due to gravity).
• Mass on a Spring: T = 2π * √(M/K), where:
  • T is the time period.
  • M is the mass on the spring.
  • K is the spring constant.

Pendulum (Ball on a String) Analysis

• Mass Independence: The time period of a pendulum is independent of the mass of the bob. A golf ball and a car (ideally, in a vacuum) would have the same time period if suspended from the same length string.
• Amplitude Independence: The time period is also independent of the amplitude (displacement). This is why pendulums are used in clocks; the decreasing amplitude doesn't significantly affect the timekeeping.
• Gravitational Dependence: The time period does depend on the gravitational field strength (G). Therefore, a pendulum's time period would change on a different planet.

Mass on a Spring Analysis

• Mass Dependence: The time period of a mass on a spring does depend on the mass attached.
• Gravitational Independence: The time period is independent of the gravitational field strength (G). The system will oscillate with the same time period even in space where G is zero.
• Amplitude Independence: Similar to the pendulum, the time period is independent of the amplitude (how far the mass is pulled down).

Proportionality Analysis

The video emphasizes understanding the proportional relationships within the time period equations.

• Pendulum:
  • T ∝ √(L/G)
  • If G is constant: T ∝ √L
  • If L is constant: T ∝ 1/√G
• Example 1: If the length of the pendulum doubles (L x 2), the time period increases by a factor of √2.
• Example 2: If G gets weaker by a factor of four (G / 4), the time period increases by a factor of two.
• Mass on a Spring:
  • T ∝ √M
  • T ∝ 1/√K
• Example: If a spring is four times stiffer (K x 4), the time period is halved (T / 2).

Experimental Determination of G and K

The video explains how to experimentally determine G (gravitational field strength) using a pendulum and K (spring constant) using a mass-spring system.

• Finding G:
  1. Rearrange the pendulum equation to isolate G: G = 4π²L/T².
  2. Plot a graph of L (length) vs. T² (time period squared).
  3. The gradient of the graph is equal to G/4π², so G = 4π² * gradient.
• Finding K:
  1. Rearrange the mass-spring equation to isolate K: K = 4π²M/T².
  2. Plot a graph of M (mass) vs. T² (time period squared).
  3. The gradient of the graph is equal to K/4π², so K = 4π² * gradient.

Experimental Procedure and Error Reduction

• Multiple Readings: Take multiple readings (10-20 oscillations) to improve accuracy.
• Fiducial Marker: Use a fiducial marker (a straight line at the equilibrium point) to accurately count oscillations.
• Timing Technique: Start the timer when the pendulum or mass passes the fiducial marker, counting "zero" at the start and then counting oscillations (1, 2, 3,...10).
• Time Period Calculation: Divide the total time for 10 oscillations by 10 to find the time period for a single oscillation. This reduces uncertainties due to human reaction time.

Conclusion

The video provides a detailed explanation of simple harmonic motion using the examples of a pendulum and a mass on a spring. It covers the relevant equations, proportionalities, and experimental methods for determining key physical constants. The emphasis on experimental technique and error reduction makes the information practical and applicable.