The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the

Q: Why do Torque and Work have the same dimensions if they are completely different physical concepts?

Dimensions only tell us about the fundamental base quantities involved (Force $\times$ Length), not the vector nature or the physical context. Torque is a cross product (vector), while work is a dot product (scalar), but their base units are identical.

Q: Is a Light Year a unit of time?

No, a Light Year is the distance that light travels in a vacuum in one Julian year. Since it measures distance, its dimensional formula is simply $[\mathrm{L}]$.

Q: Why doesn't strain affect the dimensions of Young's Modulus?

Strain is defined as the ratio of change in dimension to the original dimension (e.g., $\Delta L / L$). Since it is a ratio of two identical quantities, it is dimensionless. Thus, Young's Modulus has the same dimensions as Stress.

Q: How do I quickly find the dimension of Angular Momentum?

Use the formula $L = mvr$. Mass is $[\mathrm{M}]$, velocity is $[\mathrm{L T^{-1}}]$, and radius is $[\mathrm{L}]$. Multiplying them gives $[\mathrm{M L^2 T^{-1}}]$.

Q: Can two quantities with the same dimensions be added together?

Yes, but only if they represent the same physical nature. You can add two energies, but you cannot physically add Torque and Work, even though their dimensions match, because they represent different physical phenomena.

Question

The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair (s).

Options:

(A) Torque and work

(B) Angular momentum and work

(C) Energy and Young's modulus

(D) Light year and wavelength

Sigma Percentile · Accepted Answer

### The Magic of Dimensional Analysis

Dimensional analysis is one of the most powerful tools in a physicist's arsenal. It allows us to check the consistency of equations, derive relationships between physical quantities, and, as in this classic problem, uncover hidden symmetries between seemingly unrelated concepts.

Our mission here is to evaluate four pairs of physical quantities and determine which pairs share the exact same dimensional formula. Remember, two quantities can have identical dimensions even if they describe entirely different physical phenomena.

### Analyzing Torque and Work

Let's begin with the first pair: **Torque** and **Work**.

Torque ($	au$) is the rotational analog of force, representing the turning effect of a force applied at a distance from an axis. Mathematically, it is the cross product of the position vector and the force vector. Its magnitude is given by Force $	imes$ Perpendicular Distance.

Work ($W$), on the other hand, is the energy transferred when a force acts upon an object over a displacement. It is the dot product of the force vector and the displacement vector.

Despite one being a vector (Torque) and the other a scalar (Work), both are fundamentally calculated by multiplying a force by a length. Therefore, their dimensional formulas are identical:

$$[	ext{Torque}] = [	ext{Force}] 	imes [	ext{Length}] = [\mathrm{M L T^{-2}}] 	imes [\mathrm{L}] = [\mathrm{M L^2 T^{-2}}]$$

$$[	ext{Work}] = [	ext{Force}] 	imes [	ext{Length}] = [\mathrm{M L T^{-2}}] 	imes [\mathrm{L}] = [\mathrm{M L^2 T^{-2}}]$$

Thus, Torque and Work are a matching pair!

### The Case of Angular Momentum

Next, we examine **Angular Momentum** and **Work**.

We already established that the dimension of Work is $[\mathrm{M L^2 T^{-2}}]$.

Angular momentum ($L$) for a point particle is defined as the cross product of its position vector and its linear momentum ($p = mv$).

$$[	ext{Angular Momentum}] = [	ext{Mass}] 	imes [	ext{Velocity}] 	imes [	ext{Radius}]$$

$$[	ext{Angular Momentum}] = [\mathrm{M}] 	imes [\mathrm{L T^{-1}}] 	imes [\mathrm{L}] = [\mathrm{M L^2 T^{-1}}]$$

Comparing $[\mathrm{M L^2 T^{-1}}]$ with $[\mathrm{M L^2 T^{-2}}]$, we can clearly see that they do not match. The time dimension differs by a power of one.

### Energy vs. Young's Modulus

Our third pair is **Energy** and **Young's Modulus**.

Energy, being the capacity to do work, shares the exact same dimensional formula as Work: $[\mathrm{M L^2 T^{-2}}]$.

Young's Modulus ($Y$) is a measure of the stiffness of a solid material. It is defined as the ratio of tensile stress to tensile strain.

$$Y = \frac{	ext{Stress}}{	ext{Strain}}$$

Strain is the ratio of change in length to original length ($\Delta L / L$), making it a **dimensionless** quantity. Therefore, the dimensions of Young's Modulus are entirely determined by Stress, which is Force per unit Area.

$$[	ext{Young's Modulus}] = \frac{[	ext{Force}]}{[	ext{Area}]} = \frac{[\mathrm{M L T^{-2}}]}{[\mathrm{L^2}]} = [\mathrm{M L^{-1} T^{-2}}]$$

Clearly, $[\mathrm{M L^2 T^{-2}}]$ and $[\mathrm{M L^{-1} T^{-2}}]$ are not a match.

### The Light Year Trap

Finally, we look at **Light Year** and **Wavelength**.

The term "Light Year" often tricks students into thinking it is a unit of time because of the word "year". However, a light year is defined as the **distance** that light travels in a vacuum in one Julian year. Since it is a measure of distance, its dimensional formula is simply $[\mathrm{L}]$.

Wavelength, as the name implies, is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is also a measure of length, so its dimensional formula is $[\mathrm{L}]$.

Both quantities share the exact same dimension, making them a perfect match!

### The Final Verdict

By systematically breaking down each physical quantity into its fundamental dimensions, we successfully identified the matching pairs. Torque and Work share the dimensions of $[\mathrm{M L^2 T^{-2}}]$, while Light Year and Wavelength share the dimension of $[\mathrm{L}]$.

This makes options (a) and (d) the correct answers to this multiple-choice question.

Animated Solution for Physics - Physics and Measurement: The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair (s).

Select Answer:

Visualized Solution

$Dimensional Analysis$

$Option (a): Torque and Work$

$Option (b): Angular Momentum and Work$

$Option (c): Energy and Young’s Modulus$

$Option (d): Light Year and Wavelength$

$Conclusion$

The Sigma Insight: Dimensional Analysis

The Magic of Dimensional Analysis

Analyzing Torque and Work

The Case of Angular Momentum

Energy vs

The Light Year Trap

The Final Verdict

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