Q.Which of the following pairs of physical quantities does not have same dimensional formula?

The key idea here is Dimensional Analysis, which involves comparing the fundamental dimensions (Mass $[M]$, Length $[L]$, Time $[T]$) of physical quantities. Quantities with the same physical nature must have the same dimensional formula.

1. Work ($W = F \cdot d$) has dimensions $[M L T^{-2}] \cdot [L] = [M L^2 T^{-2}]$. Torque ($\tau = r \times F$) also has dimensions $[L] \cdot [M L T^{-2}] = [M L^2 T^{-2}]$. So, (A) has the same dimensions.
2. Angular momentum ($L = I \omega = (M R^2) (T^{-1})$) has dimensions $[M L^2 T^{-1}]$. Planck's constant ($h = E/f = (M L^2 T^{-2}) / (T^{-1})$) also has dimensions $[M L^2 T^{-1}]$. So, (B) has the same dimensions.
3. Tension is a force, so its dimensions are $[M L T^{-2}]$. Surface tension ($\gamma = F/L$) has dimensions $[M L T^{-2}] / [L] = [M T^{-2}]$. These are different. …

Understanding the fundamental dimensions of physical quantities is crucial in physics. Dimensional analysis allows us to check the consistency of equations, derive relationships between physical quantities, and even predict the nature of unknown quantities. Every physical quantity can be expressed in terms of fundamental dimensions: Mass (M), Length (L), and Time (T).

When two physical quantities have the same dimensional formula, it implies they are fundamentally similar in terms of their physical nature, even if they represent different concepts or are measured in different units. For example, work and energy both represent the capacity to do work and share the same dimensions. If two quantities have different dimensional formulas, they are fundamentally different and cannot be equated or added together.

Let's systematically determine the dimensional formula for each quantity in the given pairs:

1. Recall Fundamental Dimensions:

   We use the following basic dimensions:

   • Mass: $[M]$
   • Length: $[L]$
   • Time: $[T]$

2. Analyze Option (A): Work and Torque

   • Work ($W$): Work is defined as force multiplied by displacement.

     • First, let's find the dimensions of Force ($F$). Force is mass times acceleration ($F = ma$).
       • $[m] = [M]$
       • $[a] = [L][T]^{-2}$ (acceleration is change in velocity per unit time, and velocity is displacement per unit time)
       • $[F] = [M][L][T]^{-2}$
     • Now, for Work:
       • $[W] = [F] \times [\text{displacement}]$
       • $[W] = ([M][L][T]^{-2}) \times [L]$
       • $[W] = [M][L]^2[T]^{-2}$

   • Torque ($\tau$): Torque is defined as force multiplied by the perpendicular distance from the axis of rotation.

     • $[\tau] = [F] \times [\text{distance}]$
     • $[\tau] = ([M][L][T]^{-2}) \times [L]$
     • $[\tau] = [M][L]^2[T]^{-2}$

   • Comparison: Both Work and Torque have the dimensional formula $[M][L]^2[T]^{-2}$. So, this pair has the same dimensional formula.

3. Analyze Option (B): Angular Momentum and Planck's Constant

   • Angular Momentum ($L$): Angular momentum can be defined as the product of linear momentum and the perpendicular distance from the axis of rotation, or $L = mvr$.

     • $[m] = [M]$
     • $[v] = [L][T]^{-1}$ (velocity is displacement per unit time)
     • $[r] = [L]$
     • $[L] = [M] \times ([L][T]^{-1}) \times [L]$
     • $[L] = [M][L]^2[T]^{-1}$

   • Planck's Constant ($h$): Planck's constant relates the energy of a photon ($E$) to its frequency ($\nu$) via the equation $E = h\nu$.

     • From this, $h = E/\nu$.
     • We already found the dimensions of Energy (which is the same as Work) in step 2: $[E] = [M][L]^2[T]^{-2}$.
     • Frequency ($\nu$) is the reciprocal of time period ($1/T$).
       • $[\nu] = [T]^{-1}$
     • Now, for Planck's Constant:
       • $[h] = [E] / [\nu]$
       • $[h] = ([M][L]^2[T]^{-2}) / ([T]^{-1})$
       • $[h] = [M][L]^2[T]^{-2}[T]^1$
       • $[h] = [M][L]^2[T]^{-1}$

   • Comparison: Both Angular Momentum and Planck's Constant have the dimensional formula $[M][L]^2[T]^{-1}$. So, this pair has the same dimensional formula.

4. Analyze Option (C): Tension and Surface Tension

   • Tension ($T$): Tension is a type of force, specifically the pulling force transmitted axially by means of a string, cable, chain, or similar one-dimensional continuous object.

     • As determined in step 2, the dimensions of Force are:
       • $[\text{Tension}] = [F] = [M][L][T]^{-2}$

   • Surface Tension ($\gamma$): Surface tension is defined as the force per unit length acting perpendicular to a line drawn on the surface of a liquid. …

Concept: Dimensional Analysis

Dimensional analysis compares the fundamental dimensions (mass $M$, length $L$, time $T$, etc.) of physical quantities. Two quantities have the same dimensional formula if they can be expressed in identical powers of $M$, $L$, $T$, etc.

Method: Derive and Compare Dimensions

Step 1: Write the defining formula for each quantity in terms of fundamental units.

Step 2: Replace each physical quantity with its dimension $[M^a L^b T^c]$.

Step 3: Simplify to get the dimensional formula.

Step 4: Compare the two quantities in each pair.

Pair (A): Work and Torque

• Work = Force × Displacement

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

• Torque = Force × Perpendicular distance

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

✓ Same dimensional formula.

Pair (B): Angular Momentum and Planck’s Constant

• Angular momentum = Moment of inertia × Angular velocity

  $[Angular\ momentum] = [M L^2] \times [T^{-1}] = [M L^2 T^{-1}]$

• Planck’s constant = Energy / Frequency

  $[h] = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}]$

✓ Same dimensional formula.

Pair (C): Tension and Surface Tension

• Tension = Force (in a string/rope)

  $[Tension] = [M L T^{-2}]$

• Surface tension = Force / Length …

Let’s break this down step-by-step — first the concept, then the common mistakes, and finally how to avoid each.

Concept: Dimensional Analysis

Every physical quantity can be expressed in terms of fundamental dimensions:

Mass ($M$), Length ($L$), Time ($T$), etc.

Two quantities have the same dimensional formula if their dependence on $M$, $L$, $T$ (and others) is identical.

Step 1: Check each pair

(A) Work and Torque

• Work = Force × displacement → $[M L T^{-2}] \times [L] = [M L^2 T^{-2}]$
• Torque = Force × perpendicular distance → $[M L T^{-2}] \times [L] = [M L^2 T^{-2}]$

✓ Same dimensions

(B) Angular momentum and Planck’s constant

• Angular momentum = $m v r$ → $[M] \times [L T^{-1}] \times [L] = [M L^2 T^{-1}]$
• Planck’s constant ($h$) = Energy / frequency → $\frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}]$

✓ Same dimensions

(C) Tension and Surface tension

• Tension (force) → $[M L T^{-2}]$
• Surface tension = Force / length → $\frac{[M L T^{-2}]}{[L]} = [M T^{-2}]$

✗ Different dimensions

(D) Impulse and Linear momentum

• Impulse = Force × time → $[M L T^{-2}] \times [T] = [M L T^{-1}]$
• Linear momentum = mass × velocity → $[M] \times [L T^{-1}] = [M L T^{-1}]$

✓ Same dimensions

Answer: (C) Tension and surface tension

Common Mistakes Students Make

✗ Mistake 1: Confusing “Tension” with “Surface Tension”

• Why it happens: Both have the word “tension,” so students assume they are related.
• How to avoid:
  • Tension is a force (unit: newton).
  • Surface tension is force per unit length (unit: N/m).
  • Always write the defining formula before comparing dimensions.

✗ Mistake 2: Forgetting the formula for Planck’s constant

• Why it happens: Planck’s constant is less familiar than momentum or torque.
• How to avoid:
  • Remember: $E = h \nu$ → $h = \frac{E}{\nu}$
  • Energy $[M L^2 T^{-2}]$, frequency $\nu$ has dimension $[T^{-1}]$
  • So $h$ has $[M L^2 T^{-1}]$ — same as angular momentum.

✗ Mistake 3: Thinking Work and Torque are different because they “feel” different

• Why it happens: Work is scalar, torque is vector — but dimension doesn’t care about scalar/vector.
• How to avoid: …