A dust particle oscillates in air with a time period which...

A dust particle oscillates in air with a time period which depends on atmospheric pressure $P$, density of air $d$ and energy of the particle $E$, then time period is proportional to (A) $\begin{array}{ll}P^{-\frac{5}{6}} d^{\frac{1}{2}} E^{\frac{1}{3}} & \text { (B) } P^{\frac{1}{2}} d^{3} E^{-2}\end{array}$ (C) $\begin{array}{ll}P^{-\frac{1}{3}} d^{\frac{1}{2}} E^{2} & \text { (D) } P^{-2} d^{-\frac{1}{2}} E^{-3}\end{array}$

A dust particle oscillates in air with a time period which depends on atmospheric pressure $P$, density of air $d$ and energy of the particle $E$, then time period is proportional to
(A)
$\begin{array}{ll}P^{-\frac{5}{6}} d^{\frac{1}{2}} E^{\frac{1}{3}} & \text { (B) } P^{\frac{1}{2}} d^{3} E^{-2}\end{array}$
(C)
$\begin{array}{ll}P^{-\frac{1}{3}} d^{\frac{1}{2}} E^{2} & \text { (D) } P^{-2} d^{-\frac{1}{2}} E^{-3}\end{array}$

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Key Concepts

Dimensional Analysis

Dimensional analysis involves checking and ensuring that both sides of an equation have the same dimensions, such as mass, length, and time. It is a critical tool in physics to derive relationships between physical quantities by comparing their fundamental units, without needing to know the details of the underlying mechanisms.

Scaling Laws

Scaling laws describe how physical quantities change when the size or magnitude of a system is scaled up or down. They allow us to determine how one measurable quantity is proportional to powers of other variables, which is particularly useful in problems involving proportionality such as the time period's dependence on pressure, density, and energy.

Physical Quantities and Units

The understanding of physical quantities and their corresponding units is essential when applying dimensional analysis. Pressure, density, energy, and time period each have specific dimensions. Recognizing their dimensions allows one to systematically combine them in a way that the units match on both sides of an equation, ensuring the mathematical correctness of the derived formula.

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