Units and Measurements

1. Units and Dimensions

Fundamental Units: Mass (kg), Length (m), Time (s), Current (A), Temp (K), Amount (mol), Intensity (cd).
Derived Units: Derived from fundamental (e.g., Velocity m/s, Force N).
Dimensions: Powers to which fundamental units are raised. [M^a L^b T^c].

Homogeneity Principle: Dimensions of each term in an equation must be same.
Correctness Check: LHS [Dimensions] = RHS [Dimensions].
Conversion: n₁u₁ = n₂u₂.

2. Errors in Measurement

Difference between measured and true value.

Absolute Error: |a_true - a_measured|.
Relative Error: Mean Absolute Error / Mean Value.
Percentage Error: Relative Error × 100%.
Propagation:
Sum/Diff: Add absolute errors (ΔA + ΔB).
Product/Div: Add percentage errors (%A + %B).
Power (Aⁿ): n × %A.

(Asked in NEET 2018, 2020, 2022)

3. Measuring Instruments

Instruments for precise length measurement.

Vernier Caliper:
Least Count (LC) = 1 MSD - 1 VSD.
Reading = MSR + (VSR × LC).
Screw Gauge:
LC = Pitch / Total Circular Divisions.
Reading = Main Scale + (Circular Scale × LC).

(Asked in NEET 2016, 2019, 2021, 2023)

4. Significant Figures

Rules for counting SF:
1. All non-zeros are significant.
2. Zeros between non-zeros are significant.
3. Leading zeros are NOT significant.
4. Trailing zeros in number with decimal are significant.
Rounding Off:
If digit > 5, increment previous.
If digit = 5 and previous is odd, increment. If even, leave.

Numericals: Units & Dimensions

Q1. A physical quantity X = a² b³ / c√d. % errors in a, b, c, d are 2%, 1%, 3%, 4%. Find max % error in X.

Solution:
%X = 2(%a) + 3(%b) + 1(%c) + 0.5(%d).
%X = 2(2) + 3(1) + 1(3) + 0.5(4).
%X = 4 + 3 + 3 + 2 = 12%.

Q2. Find dimensions of a/b in equation P = (a - t²) / bx. (P=Pressure, x=dist, t=time).

Solution:
1. [a] must match [t²]. So [a] = [T²].
2. [P] = [a] / ([b][x]).
[b] = [a] / ([P][x]) = [T²] / ([ML^-1T^-2] [L]).
[b] = [T²] / [MT^-2] = [M^-1T⁴].
3. Ratio [a/b] = [T²] / [M^-1T⁴] = [M T^-2].

Q3. In a VC, 10 VSD = 9 MSD. 1 MSD = 1mm. Zero error is -0.02 cm. Main scale reads 2.4 cm, 6th division coincides. Diameter?

Solution:
LC = 1 MSD - 0.9 MSD = 0.1 MSD = 0.1 mm = 0.01 cm.
Reading = MSR + (VSR × LC) - Zero Error.
R = 2.4 + (6 × 0.01) - (-0.02).
R = 2.4 + 0.06 + 0.02 = 2.48 cm.

Q4. Pitch 1mm, 100 circular divisions. Main scale read 2mm, 45th div coincides. No zero error. Thickness?

Solution:
LC = Pitch / Divs = 1 mm / 100 = 0.01 mm.
Reading = MS + (CS × LC).
R = 2 mm + (45 × 0.01mm) = 2 + 0.45 = 2.45 mm.

Q5. Calculate area of rectangle L=2.5 m, B=1.25 m to correct significant figures.

Solution:
Area = 2.5 × 1.25 = 3.125 m².
Rule: Result must have same SF as term with least SF.
L has 2 SF. B has 3 SF. Result needs 2 SF.
Round 3.125 to 2 SF → 3.1 m².

Q6. If Force F, Velocity V, Time T are fundamental, determine dimensions of Mass.

Solution:
Force = Mass × Accel = M (V/T).
M = F T / V.
Dimensions = [F V^-1 T].

Q7. Error in measuring radius of sphere is 2%. Error in volume?

Solution:
V = (4/3)πr³.
%V = 3 × %r.
%V = 3 × 2% = 6%.

Q8. Check dimensional consistency: KE = (3/16)mv².

Solution:
LHS: Energy [ML²T^-2].
RHS: Mass × Velocity² = [M] [LT^-1]² = [ML²T^-2].
Dimensions match. Equation is dimensionally consistent (but numerically wrong as factor is 1/2, dimensions can't catch constant errors).

Q9. Add 2.45 g, 3.2 g, 4.123 g with correct SF.

Solution:
Sum = 2.45 + 3.2 + 4.123 = 9.773 g.
Least decimal places is 1 (in 3.2).
Round result to 1 decimal place: 9.773 → 9.8 g.

Q10. Find dimensions of η in F = 6πηrv.

Solution:
η = F / (r v).
[η] = [MLT^-2] / ([L] [LT^-1]).
[η] = [MLT^-2] / [L²T^-1] = [ML^-1T^-1].

Important Formulae

1. Dimensional Analysis

Common Dimensions:

Force: [MLT^-2] | Energy: [ML²T^-2]

Power: [ML²T^-3] | Pressure: [ML^-1T^-2]

Conversion: n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c

2. Error Analysis

Percentage Error:

For Z = A^p B^q / C^r

%Z = p(%A) + q(%B) + r(%C)

Errors always add up (Max possible error).

3. Least Count

Vernier Caliper:

LC = 1 MSD - 1 VSD

Screw Gauge:

LC = Pitch / No. of Circular Divs

4. Important Constants

G (Gravitational): [M^-1L³T^-2]

h (Planck's): [ML²T^-1]

η (Viscosity): [ML^-1T^-1]

20 NEET Golden Facts

1. Same Dimensions 1: Work, Energy, Torque, Heat all have [ML²T^-2].
2. Same Dimensions 2: Impulse and Momentum have [MLT^-1].
3. Same Dimensions 3: Pressure, Stress, Modulus of Elasticity, Energy Density [ML^-1T^-2].
4. Dimensionless: Strain, Refractive Index, Relative Density, Angle (Plane & Solid), Efficiency.
5. Unit but no Dimension: Angle (Radian), Solid Angle (Steradian).
6. Least Count Error: Minimized by taking multiple observations or using better instrument.
7. Zero Error: Positive if zero of vernier is right of main zero. Negative if left. Correct by subtracting Z.E. with sign.
8. Significant Figure Sum: Result should have same decimal places as the number with least decimal places.
9. Significant Figure Product: Result should have same SF as number with least SF.
10. 1 Light Year: Distance light travels in 1 year. Unit of Length. 9.46 × 10¹⁵ m.
11. 1 Parsec: 3.26 Light Years. Largest practical unit of distance.
12. Error in g: If % error in length `l` and time `t` are given, max error in `g` from pendulum (T=2π√(l/g)) is %l + 2(%t).
13. Angular Momentum (L): Equivalent dimension to Planck's Constant (h). [ML²T^-1].
14. RC, L/R, √(LC): All have dimensions of Time [T].
15. Systematic Error: Errors in one direction (either positive or negative). e.g., Zero error, Instrument defect.
16. Random Error: Irregular errors. Reduced by taking mean of n observations (Error reduces by 1/√n or 1/n depending on definition, mean error reduces by 1/n).
17. 1 AMU: 1.66 × 10^-27 kg. Atomic Mass Unit.
18. Solar Constant: Energy/Area/Time. Dimension [ML⁰T^-3]. Same as Intensity.
19. Surface Tension (T): Force/Length. [MT^-2]. Same as Spring Constant (k).
20. Frequency (f): Dimension [T^-1]. Same as Angular Velocity ω and Velocity Gradient.

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