A tiny metallic rectangular sheet has length and breadth of 5 ~mm$5 \mathrm{~mm}$ and 2.5 ~mm$2.5 \mathrm{~mm}$ , respectively. Using a specially designed screw gauge which has pitch of 0.75 ~mm$0.75 \mathrm{~mm}$ and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be x100$\frac{\mathrm{x}}{100}$ where x$\mathrm{x}$ is ________.

Core Logic

Let's derive the dimensional formulas systematically:

• (A) Mass density: ρ = MassVolume = (M)/(L³) = [M¹ L⁻³ T⁰] (IV)$\rho = \frac{\text{Mass}}{\text{Volume}} = \frac{M}{L^3} = [M^1 L^{-3} T^0] \implies \text{(IV)}$ [cite: 765].
• (B) Impulse: I = F · Δ t = [M¹ L¹ T⁻²] · [T] = [M¹ L¹ T⁻¹] (II)$I = F \cdot \Delta t = [M^1 L^1 T^{-2}] \cdot [T] = [M^1 L^1 T^{-1}] \implies \text{(II)}$ [cite: 767].
• (C) Power: P = WorkTime = [M¹ L² T⁻²][T] = [M¹ L² T⁻³] (I)$P = \frac{\text{Work}}{\text{Time}} = \frac{[M^1 L^2 T^{-2}]}{[T]} = [M^1 L^2 T^{-3}] \implies \text{(I)}$ [cite: 770].
• (D) Moment of inertia: I = M r² = [M¹ L² T⁰] (III)$I = M r^2 = [M^1 L^2 T^0] \implies \text{(III)}$ [cite: 772].

Matching all four pairings establishes the layout: (A)-(IV), (B)-(II), (C)-(I), (D)-(III)[cite: 110, 758].

Pattern Recognition

Isolating the unique matching option for a straightforward parameter like Mass Density (A-IV) or Moment of Inertia (D-III) easily allows exclusion of multiple invalid answer branches instantly[cite: 765, 772].

Chapter Mix

Class 11 Physics: Units and Measurements

Related Formula

The Least Count (LC) of a screw gauge tool is defined by:

Core Logic

The initial least count is given as:

Now, calculate the modified parameters from the text details :

• New Pitch: P' = P(1 + 0.75) = 1.75P$P' = P(1 + 0.75) = 1.75P$
• New Divisions: N' = N(1 - 0.50) = 0.5N$N' = N(1 - 0.50) = 0.5N$

Step 1: Calculating the New Least Count

Set up the updated least count expression ratio :

Substitute the initial least count value :

Converting into the requested scientific prefix units (10⁻³ mm$10^{-3}\text{ mm}$) :

Therefore, the requested value is 35.

Pattern Recognition

Least count scales proportionally with pitch increases, and inversely with reductions in circular divisions.

Chapter Mix

Class 11 Physics: Units and Measurements

Related Formula

In multiplication and division arithmetic rules, the final product or quotient must be rounded off to retain as many significant figures as are present in the least precise operand.

Core Logic

Let us check the significant digit count of the operands in the expression :

• 32.3$32.3$ has 3$3$ significant figures.
• 1125$1125$ has 4$4$ significant figures.
• 27.4$27.4$ has 3$3$ significant figures.

The minimum number of significant figures among the numbers is 3$3$.

Step 1: Rounding Off

Direct calculation yield :

y = 1326.186...$y = 1326.186...$

Rounding this value off to contain exactly 3 significant figures means changing it to 1330, since the digit after 2 is 6 (which is greater than 5), updating the hundreds spot upwards.

Pattern Recognition

Never keep unearned precision from automated calculation. The output is bounded strictly by your least precise entry.

Chapter Mix

Class 11 Physics: Units and Measurements

Related Formula

• Angular Impulse = Change in Angular Momentum = τ · Δ t = [M L² T⁻²] · [T] = [M L² T⁻¹]$\tau \cdot \Delta t = [M L^2 T^{-2}] \cdot [T] = [M L^2 T^{-1}]$ [cite: 670, 672]
• Latent Heat (L$L$) = (Q)/(m) = [M L² T⁻²][M] = [M⁰ L² T⁻²]$\frac{Q}{m} = \frac{[M L^2 T^{-2}]}{[M]} = [M^0 L^2 T^{-2}]$
• Electrical resistivity (\rho) = (R · A)/(l) = [M L² T⁻³ A⁻²] · [L²][L] = [M L³ T⁻³ A⁻²]$\frac{R \cdot A}{l} = \frac{[M L^2 T^{-3} A^{-2}] \cdot [L^2]}{[L]} = [M L^3 T^{-3} A^{-2}]$
• Electromotive force (V$V$) = (W)/(q) = [M L² T⁻²][A T] = [M L² T⁻³ A⁻¹]$\frac{W}{q} = \frac{[M L^2 T^{-2}]}{[A T]} = [M L^2 T^{-3} A^{-1}]$

Core Logic

By comparing the formulas derived for each physical quantity with the options given in List-II [cite: 670, 673, 674]:

• (A) matches with (III) [cite: 670, 672]
• (B) matches with (I)
• (C) matches with (IV)
• (D) matches with (II)

Hence, the correct matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).

Pattern Recognition

In match-the-column dimensional analysis questions, identifying even one or two straightforward quantities like Latent Heat (L = Q/m$L = Q/m$) often immediately eliminates three incorrect options, securing a quick correct answer.

Chapter Mix

Class 11 Physics: Units and Measurements

Core Logic

Let\'s check the dimensions of each pair :

• Torque = [Energy] = [ML²T⁻²]$\text{Torque} = [\text{Energy}] = [ML^2T^{-2}]$
• Surface Tension = [MT⁻²]$\text{Surface Tension} = [MT^{-2}]$ vs Impulse = [MLT⁻¹]$\text{Impulse} = [MLT^{-1}]$
• [Angular Momentum] = [Planck's Constant] = [ML²T⁻¹]$[\text{Angular Momentum}] = [\text{Planck\'s Constant}] = [ML^2T^{-1}]$
• [Pressure] = [Young's Modulus] = [ML⁻¹T⁻²]$[\text{Pressure}] = [\text{Young\'s Modulus}] = [ML^{-1}T^{-2}]$

Step 1: Identify Non-Matching Pair

Surface tension and impulse do not share matching dimension frameworks.

Pattern Recognition

Surface tension is force per unit length ([MT⁻²]$[MT^{-2}]$), while impulse is force times time ([MLT⁻¹]$[MLT^{-1}]$)

Chapter Mix

Class 11 Physics: Units and Measurements