Chapter 1: Problem 78
The period \(T\) of a simple pendulum is the amount of time required for it to undergo one complete oscillation. If the length of the pendulum is \(L\) and the acceleration due to gravity is \(g\), then the period is given by $$ T=2 \pi L^{p} g^{q} $$ Find the powers \(p\) and \(q\) required for dimensional consistency.
Short Answer
Step by step solution
Understand the formula components
Write dimensions for each component
Formulate the dimensional equation
Simplify the dimensional equation
Equate dimensions for consistency
Solve for p and q
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Pendulum Period Formula
- Describing the simple harmonic motion in clocks.
- Predicting changes in environments with varying gravity.
- Observing how adjusting the pendulum’s length can change clock time intervals.
Dimensional Consistency
- Ensure the dimensions of \( T \) on the left side are equal to the dimensions on the right side containing elements with \( L \) and \( g \).
- Express each component in terms of fundamental dimensions, here \( L \) for length, and \( T \) for time.
Pendulum Oscillation
- The pendulum's period is independent of its total mass.
- It remains consistent for small angle swings (simple harmonic approximation).
- It exhibits periodic motion, which repeats in equal intervals of time.
- Used in calibrating time in clocks, providing precision and reliability.
- Helps physicists understand gravitational fields by measuring time periods.
- Represents a perfect model for studying frictionless motion and energy conservation in systems.