Understanding Physical Dimensions

In physics, the dimension of a physical quantity tells us how it is related to the fundamental base quantities like mass (M), length (L), and time (T). Two quantities can only be added, subtracted, or equated if they have the same dimensions. Let's find the dimensions of linear momentum and compare it with the dimensions of the given options.

Dimension of Linear Momentum

Linear momentum (\(p\)) is defined as the product of mass (\(m\)) and velocity (\(v\)).

\[p = m \times v\]

The dimensions are:

• Dimension of mass (\(m\)) is [M].
• Dimension of velocity (\(v\)) is [L][T]\(^{-1}\) (since velocity is displacement per unit time).

So, the dimension of linear momentum is:

\[[p] = [M] \times [L][T]^{-1} = [M][L][T]^{-1}\]

The dimension of linear momentum is [M][L][T]\(^{-1}\).

Analyzing Dimensions of Options

Now let's find the dimensions of each given option:

1. Impulse Dimension

Impulse (\(J\)) is defined as the product of force (\(F\)) and the time interval (\(\Delta t\)) over which the force acts.

\[J = F \times \Delta t\]

First, let's find the dimension of force. Force is mass times acceleration (\(F = ma\)).

• Dimension of mass (\(m\)) is [M].
• Dimension of acceleration (\(a\)) is [L][T]\(^{-2}\) (since acceleration is change in velocity per unit time).

So, the dimension of force (\(F\)) is:

\[[F] = [M] \times [L][T]^{-2} = [M][L][T]^{-2}\]

Now, the dimension of impulse is:

\[[J] = [F] \times [\Delta t] = [M][L][T]^{-2} \times [T] = [M][L][T]^{-1}\]

The dimension of impulse is [M][L][T]\(^{-1}\). This matches the dimension of linear momentum.

Physically, the Impulse-Momentum Theorem states that the impulse acting on an object is equal to the change in its linear momentum (\(\Delta p = J\)). Since impulse is equal to a change in linear momentum, they must have the same dimensions.

2. Work Dimension

Work (\(W\)) is defined as the product of force (\(F\)) and displacement (\(d\)) in the direction of the force.

\[W = F \times d\]

We already know the dimension of force is [M][L][T]\(^{-2}\).

• Dimension of force (\(F\)) is [M][L][T]\(^{-2}\).
• Dimension of displacement (\(d\)) is [L].

So, the dimension of work is:

\[[W] = [F] \times [d] = [M][L][T]^{-2} \times [L] = [M][L]^2[T]^{-2}\]

The dimension of work is [M][L]\(^2\)[T]\(^{-2}\), which is different from the dimension of linear momentum.

3. Stress Dimension

Stress (\(\sigma\)) is defined as force (\(F\)) per unit area (\(A\)).

\[\sigma = \frac{F}{A}\]

We know the dimension of force is [M][L][T]\(^{-2}\).

• Dimension of force (\(F\)) is [M][L][T]\(^{-2}\).
• Dimension of area (\(A\)) is [L]\(^2\) (since area is length times width).

So, the dimension of stress is:

\[[\sigma] = \frac{[F]}{[A]} = \frac{[M][L][T]^{-2}}{[L]^2} = [M][L]^{-1}[T]^{-2}\]

The dimension of stress is [M][L]\(^{-1}\)[T]\(^{-2}\), which is different from the dimension of linear momentum.

4. Energy Dimension

Energy comes in many forms (like kinetic energy, potential energy), but they all have the same dimension as work. Let's check the dimension of kinetic energy (\(KE\)).

\[KE = \frac{1}{2}mv^2\]

Constants like \(1/2\) are dimensionless.

• Dimension of mass (\(m\)) is [M].
• Dimension of velocity (\(v\)) is [L][T]\(^{-1}\).

So, the dimension of kinetic energy is:

\[[KE] = [M] \times ([L][T]^{-1})^2 = [M][L]^2[T]^{-2}\]

The dimension of energy is [M][L]\(^2\)[T]\(^{-2}\), which is the same as work and different from the dimension of linear momentum.

Comparison of Dimensions

Let's summarize the dimensions we found:

From the table, it is clear that only Impulse has the same dimension as linear momentum, which is [M][L][T]\(^{-1}\).

Conclusion on Dimensions

Based on the dimensional analysis, the physical quantity that has the same dimension as linear momentum is Impulse. This aligns with the Impulse-Momentum Theorem in physics, which states that impulse equals the change in momentum.

Revision Table: Key Dimensions and Formulas

Additional Information on Dimensional Analysis

Dimensional analysis is a powerful tool in physics. It can be used for:

• Checking the consistency of equations: If an equation is physically correct, the dimensions on both sides of the equation must be the same.
• Deriving relationships between physical quantities: In some cases, you can use dimensional analysis to predict how quantities might relate to each other, up to a dimensionless constant.
• Converting units: Dimensions help ensure you are converting units correctly.

Understanding the dimensions of different physical quantities like linear momentum and impulse is fundamental in mechanics. The fact that impulse and linear momentum share the same dimensions is a direct consequence of the fundamental principles relating force, time, and motion.