Understanding Dimensional Formulas in Physics

In physics, a dimensional formula is an expression that shows how fundamental quantities (like mass, length, and time) are related to a derived quantity. It is written by raising the symbols for the fundamental dimensions (M for mass, L for length, T for time, etc.) to various powers. Understanding the dimensional formula helps us check the consistency of equations and understand the nature of physical quantities.

Deriving the Dimensional Formula for Force

Force is a fundamental concept in physics that describes the interaction between objects, causing a change in their motion. According to Newton's second law of motion, force is directly proportional to the mass of an object and the acceleration it experiences.

The formula relating force, mass, and acceleration is:

$\qquad F = ma$

Where:

• $F$ is force
• $m$ is mass
• $a$ is acceleration

To find the dimensional formula of force, we need to know the dimensional formulas of mass and acceleration.

• The dimensional formula for mass ($m$) is simply $[M]$.
• Acceleration ($a$) is defined as the rate of change of velocity. Velocity is the rate of change of displacement (length).

Let's find the dimensional formula for velocity first:

$\qquad \text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$

The dimensional formula for displacement (which is a length) is $[L]$. The dimensional formula for time is $[T]$.

So, the dimensional formula for velocity is:

$\qquad [\text{Velocity}] = \frac{[L]}{[T]} = [LT^{-1}]$

Now, let's find the dimensional formula for acceleration:

$\qquad \text{Acceleration} = \frac{\text{Velocity}}{\text{Time}}$

Using the dimensional formula for velocity $[LT^{-1}]$ and for time $[T]$:

$\qquad [\text{Acceleration}] = \frac{[LT^{-1}]}{[T]} = [LT^{-1}T^{-1}] = [LT^{-2}]$

Now we can substitute the dimensional formulas for mass and acceleration into the formula for force, $F = ma$:

$\qquad [F] = [m] \times [a]$

$\qquad [F] = [M] \times [LT^{-2}]$

Combining these, we get the dimensional formula for force:

$\qquad [F] = [MLT^{-2}]$

This means that force has dimensions of mass raised to the power of 1, length raised to the power of 1, and time raised to the power of -2.

Comparing Derived Formula with Options

Let's compare our derived dimensional formula $[MLT^{-2}]$ with the given options:

• Option 1: $[MLT^{-2}]$
• Option 2: $[ML^{-3}T^{2}]$
• Option 3: $[ML^{-2}T^{-2}]$
• Option 4: $[ML^{-1}T^{-1}]$

Our derived formula matches Option 1.

Revision Table: Fundamental Dimensions and Common Derived Quantities

Additional Information on Dimensional Analysis

Dimensional analysis is a powerful tool in physics for several reasons:

• Checking Consistency: It can be used to check the dimensional consistency of equations. Every term in an equation must have the same dimensions. This is known as the principle of homogeneity of dimensions.
• Deriving Relations: In some cases, it can help in deriving relationships between physical quantities, although it cannot determine dimensionless constants.
• Unit Conversion: It simplifies the conversion of units from one system to another.
• Understanding Quantity Nature: It helps understand the fundamental nature of a derived physical quantity in terms of mass, length, time, etc.

However, dimensional analysis has limitations. It doesn't give information about dimensionless constants, trigonometric, exponential, or logarithmic functions, and it cannot distinguish between quantities that have the same dimensions (like work and torque).