Question

Pressure \(P=\left(a t^{2} / b x\right)\) where \(x=\) distance, \(t=\) time find the dimensional formula for \(\mathrm{a} / \mathrm{b}\) (a) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-4}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-2}\) (d) \(\mathrm{M}^{+1} \mathrm{~L}^{0} \mathrm{~T}^{-2}\)

Short answer

The short answer is: The dimensional formula for \(a/b\) is \(M^1L^0T^{-2}\).

Step-by-step solution

Identify the dimensions of known variables

Pressure (P) has dimensions of force per unit area, which is [ML^(-1)T^(-2)]. Time (t) has dimensions of [T], and distance (x) has dimensions of [L].

Write the given formula in terms of dimensions

We can replace each variable in the given equation with its dimensional formula \[P = \frac{at^2}{bx}\] as \[[ML^(-1)T^(-2)] = \frac{a[T]^2}{b[L]}\]

Equate dimensions on both sides of the equation

Now, equate the dimensions on the left side of the equation with the dimensions on the right side. \[ML^(-1)T^(-2) = \frac{a}{b} \cdot T^{2} L^{-1}\]

Solve for the dimensions of a/b

To find the dimensions of a/b, we need to find the exponents of M, L, and T in terms of a/b. From the previous equation, we can see: \[\frac{a}{b} = ML^0 T^{-2}\] The dimensional formula for a/b is therefore \[M^1L^0T^{-2}\] The correct answer is (d) M^1 L^0 T^{-2}.

Key concepts

Pressure

Pressure is a fundamental concept in physics that describes the force applied over a certain area. More formally, it is defined as the amount of force exerted per unit area. The formula to calculate pressure is given by:

• Pressure (P) = Force (F) / Area (A)

This means if a large force is applied over a small area, the pressure will be high. Conversely, applying the same force over a larger area will result in lower pressure.
In terms of dimensions, pressure is represented as \[ \text{Pressure } (P) = [M L^{-1} T^{-2}] \]where:

• \( M \) is mass
• \( L \) is length
• \( T \) is time.

These dimensions indicate that pressure involves mass per unit length and is also dependent on time.

Dimensional Formula

The dimensional formula is a crucial concept used to analyze the dimensions of physical quantities. It is represented through fundamental quantities such as length (L), mass (M), and time (T), ensuring consistency across equations and calculations.
The dimensional formula provides a way to derive unit consistency in physics equations. By determining the equation's balance, scientists and engineers can ensure that the mathematical forms of their calculations are consistent.
For instance, to explore the dimensional formula in the exercise:

• The given pressure \(P\) has dimensions \([M L^{-1} T^{-2}]\).
• The goal is to find the dimensions of the expression \( a/b \).
• Using dimensional analysis, the dimensions of \( a/b \) are found to be \([M^1L^0T^{-2}]\), providing consistency within the given formula.

Physics Problem Solving

Solving physics problems like the one described involves a systematic approach to understanding the underlying principles and deriving the correct solution. The main steps include:

• Identify the problem: Recognize what physical principle or concept like pressure or dimensional analysis is being tested.
• Break the problem into steps: Address complex problems by breaking them into smaller, manageable parts, similar to solving for a/b in our problem by equating dimensional sides.
• Apply known formulas: Use established equations and relationships, like the dimensional formula for pressure and other variables.
• Check units: Ensure that all measurements are in consistent units before proceeding with calculations.
• Verify the solution: Once a solution is derived, verify it through logical reasoning and dimensional analysis.

By following these steps, physics problem-solving becomes structured, making it easier to get to the correct answer.

Units and Measurements

Units and measurements are fundamental in comparing properties accurately in physics. They provide a standard for expressing quantities, ensuring clarity and precision in scientific communication.
Units are categorized into base and derived units. For practical purposes:

• Base Units: Fundamental units like meters (m) for length, kilograms (kg) for mass, and seconds (s) for time.
• Derived Units: Combinations of base units to express other quantities, like pascal (Pa) for pressure, which is equivalent to \( \text{N/m}^2\) or \( \text{kg/m s}^2\).

Accurate measurements rely on consistent units and a clear understanding of how different physical quantities relate. This understanding is essential in fields such as engineering and physics, where dimensional consistency can ensure that designs and theories behave as expected.