Units and Measurement for DDCET

📐 DDCET Exam Prep

Units and Measurement

High-Yield Master Notes for Exam Success

1 Physical Quantities & Units

A Physical Quantity is any property of material or radiation that can be measured.

$$Q = n \times u$$

Where $n$ = Numerical Value, $u$ = Unit

Systems of Units:

• MKS (SI): Meter, Kilogram, Second (Standard)
• CGS: Centimeter, Gram, Second
• FPS: Foot, Pound, Second

[Image of SI base units chart]

2 Interconversion (MKS ↔ CGS)

We use Dimensional Analysis for conversion.

$$n_2 = n_1 \left[ \frac{M_1}{M_2} \right]^a \left[ \frac{L_1}{L_2} \right]^b \left[ \frac{T_1}{T_2} \right]^c$$

🔥 Example 1: Energy

Convert 1 Joule (SI) to Erg (CGS)

Dimension: $[ML^2T^{-2}]$

Result: $1 \text{ Joule} = 10^7 \text{ Erg}$

🔥 Example 2: Force

Convert 1 Newton (SI) to Dyne (CGS)

Dimension: $[MLT^{-2}]$

Result: $1 \text{ Newton} = 10^5 \text{ Dyne}$

3 Error Analysis

Error Type	Formula
True Value ($a_m$)	Mean of all observations
Absolute Error ($\Delta a$)	$|a_m - a_{\text{measured}}|$
Percentage Error	$(\frac{\Delta a_{\text{mean}}}{a_m}) \times 100\%$

⚡ Propagation of Errors (Golden Rules)

• Sum/Diff: Add Absolute Errors ($\Delta Z = \Delta A + \Delta B$)
• Multiply/Divide: Add Relative Errors ($\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$)
• Power ($Z=A^n$): $\frac{\Delta Z}{Z} = n \cdot \frac{\Delta A}{A}$

4 Measuring Instruments

A. Vernier Caliper

[Image of Vernier Caliper showing main scale and vernier scale]

Use: Precision length measurement ($0.1$ mm).

Least Count (LC):
$\text{LC} = 1 \text{ MSD} - 1 \text{ VSD}$
Typically $0.01 \text{ cm}$.

Total Reading: $\text{MSR} + (\text{VSR} \times \text{LC})$

B. Micrometer Screw Gauge

[Image of Micrometer Screw Gauge with labeled parts]

Use: Wire diameter / Sheet thickness ($0.01$ mm).

Least Count (LC):
$\text{LC} = \frac{\text{Pitch}}{\text{Total Circular Divs}}$
Typically $0.001 \text{ cm}$.

Total Reading: $\text{MSR} + (\text{CSR} \times \text{LC})$

Units and Measurement

Exam Success માટે મહત્વની Master Notes

1 Physical Quantities & Units

Physical Quantity એટલે દ્રવ્ય (material) નો એવો ગુણધર્મ જેને માપી (measure) શકાય.

$$Q = n \times u$$

જ્યાં $n$ = Numerical Value (મૂલ્ય), $u$ = Unit (એકમ)

Systems of Units:

• MKS (SI): Meter, Kilogram, Second (Standard Physics માટે)
• CGS: Centimeter, Gram, Second
• FPS: Foot, Pound, Second

[Image of SI base units chart]

2 Interconversion (MKS ↔ CGS)

Units ને convert કરવા માટે આપણે Dimensional Analysis નો ઉપયોગ કરીએ છીએ.

$$n_2 = n_1 \left[ \frac{M_1}{M_2} \right]^a \left[ \frac{L_1}{L_2} \right]^b \left[ \frac{T_1}{T_2} \right]^c$$

🔥 Example 1: Energy (કાર્ય/ઉર્જા)

1 Joule (SI) ને Erg (CGS) માં ફેરવો.

Dimension: $[ML^2T^{-2}]$

Result: $1 \text{ Joule} = 10^7 \text{ Erg}$

🔥 Example 2: Force (બળ)

1 Newton (SI) ને Dyne (CGS) માં ફેરવો.

Dimension: $[MLT^{-2}]$

Result: $1 \text{ Newton} = 10^5 \text{ Dyne}$

3 Error Analysis (ત્રુટિ)

કોઈ પણ measurement perfect નથી હોતું. Measured value અને True value વચ્ચેના તફાવતને Error કહેવાય છે.

Error Type	Formula / સમજ
True Value ($a_m$)	બધા observations નું Mean (સરેરાશ).
Absolute Error ($\Delta a$)	$|a_m - a_{\text{measured}}|$
Percentage Error	$(\frac{\Delta a_{\text{mean}}}{a_m}) \times 100\%$

⚡ Propagation of Errors (Golden Rules)

• Addition/Subtraction: Absolute Errors હમેશા ADD થાય. ($\Delta Z = \Delta A + \Delta B$)
• Multiplication/Division: Percentage Errors હમેશા ADD થાય.
• Power ($Z=A^n$): Power ને error સાથે multiply કરો. ($\frac{\Delta Z}{Z} = n \cdot \frac{\Delta A}{A}$)

4 Measuring Instruments (માપન સાધનો)

A. Vernier Caliper

Use: ચોકસાઈપૂર્વક લંબાઈ માપવા માટે ($0.1$ mm સુધી).

Least Count (LC):
$\text{LC} = 1 \text{ MSD} - 1 \text{ VSD}$
સામાન્ય રીતે $0.01 \text{ cm}$.

Total Reading: $\text{MSR} + (\text{VSR} \times \text{LC})$

B. Micrometer Screw Gauge

[Image of Micrometer Screw Gauge with labeled parts]

Use: Wire નો diameter અથવા sheet ની thickness માપવા.

Least Count (LC):
$\text{LC} = \frac{\text{Pitch}}{\text{Total Circular Divs}}$
સામાન્ય રીતે $0.001 \text{ cm}$.

Total Reading: $\text{MSR} + (\text{CSR} \times \text{LC})$