Vidaara.orgClass 11 · Physics
CodeVID-P11-01-DIM-01
Dimensional Analysis — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
The dimensional formula of work is:
- A.$[M^1L^1T^{-2}]$
- B.$[M^1L^2T^{-2}]$
- C.$[M^1L^2T^{-3}]$
- D.$[M^0L^2T^{-2}]$
2.
The dimensional formula of pressure is:
- A.$[M^1L^{-1}T^{-2}]$
- B.$[M^1L^1T^{-2}]$
- C.$[M^1L^{-2}T^{-2}]$
- D.$[M^0L^{-1}T^{-2}]$
3.
Which pair has the SAME dimensions?
- A.work and power
- B.work and torque
- C.force and pressure
- D.velocity and acceleration
4.
Angular frequency $\\omega$ has dimensions:
- A.$[T]$
- B.$[T^{-1}]$
- C.$[LT^{-1}]$
- D.$[M^0L^0T^0]$
5.
The dimensional formula $[M^1L^2T^{-3}]$ represents:
- A.energy
- B.force
- C.power
- D.momentum
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Write the dimensional formula of momentum.
7.
Check the dimensional consistency of $s = ut + \\tfrac{1}{2}at^2$.
8.
State two quantities that are dimensionless.
9.
Find the dimensions of the universal gas constant $R$ from $PV = nRT$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Derive the dimensional formula of the coefficient of viscosity $\\eta$ from $F = \\eta A \\dfrac{dv}{dx}$.
11.
Using dimensions, convert $1\\,\\text{joule}$ into ergs.
12.
Show that $\\tfrac{1}{2}mv^2$ and $mgh$ have the same dimensions.
13.
State three limitations of dimensional analysis.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
The frequency $f$ of a vibrating string may depend on its tension $T$, length $l$ and mass per unit length $\\mu$. Using dimensional analysis, derive the relation.
15.
Find the dimensional formula of Planck's constant $h$ from $E = h\\nu$, where $\\nu$ is frequency, and state its SI unit.
Answer Key
Section A — Multiple Choice Questions
- (B) $[M^1L^2T^{-2}]$
- (A) $[M^1L^{-1}T^{-2}]$
- (B) work and torque
- (B) $[T^{-1}]$
- (C) power
Section B — Short Answer (2 marks)
- $[M^1L^1T^{-1}]$.
- Each term has dimensions $[L]$, so it is consistent.
- Strain and refractive index (also plane angle, relative density).
- $[M^1L^2T^{-2}K^{-1}\\text{mol}^{-1}]$.
Section C — Short Answer (3 marks)
- $[\\eta] = [M^1L^{-1}T^{-1}]$.
- $1\\,\\text{J} = 10^{7}\\,\\text{erg}$.
- Both reduce to $[M^1L^2T^{-2}]$ (energy).
- Cannot find dimensionless constants; fails for trig/exp/log terms; fails with more than three dependencies.
Section D — Long Answer (5 marks)
- $f = \\dfrac{k}{l}\\sqrt{\\dfrac{T}{\\mu}}$, with $k$ a dimensionless constant.
- $[h] = [M^1L^2T^{-1}]$; SI unit J s.
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