1.0 Introduction to Measurements & Units
Physics is a science of measurement. To understand the natural world, we must quantify physical quantities like length, mass, and time. A Physical Quantity is a property of a material or system that can be quantified by measurement.
What is a Unit?
A unit is a standard quantity of the same kind with which a physical quantity is compared for its measurement. Every measurement consists of two parts:
- The Numerical Value (n): How many times the unit is contained in the quantity.
- The Unit (u): The standard used for comparison.
Physical Quantity = $n \times u$
1.1 Fundamental and Derived Units
In the International System of Units (S.I.), physical quantities are classified into two categories:
| Category | Description | Examples |
|---|---|---|
| Fundamental Units | Independent of any other units. | Mass (kg), Length (m), Time (s) |
| Derived Units | Obtained by combining fundamental units. | Speed (m/s), Force (N), Volume ($m^3$) |
Density Relationship
$$\rho = \frac{M}{V}$$
Where: $\rho$ (rho) = Density, $M$ = Mass, $V$ = Volume.
Always check the power of ten in conversions.
1 picometre ($pm$) = $10^{-12}\,m$
1 nanometre ($nm$) = $10^{-9}\,m$
1 micrometre ($\mu m$) = $10^{-6}\,m$
The density of copper is $8.9\,g/cm^3$. Express this value in S.I. units ($kg/m^3$).
Solution:
1. We know: $1\,g = 10^{-3}\,kg$ and $1\,cm^3 = 10^{-6}\,m^3$.
2. Relationship: $1\,g/cm^3 = \frac{10^{-3}\,kg}{10^{-6}\,m^3} = 10^3\,kg/m^3$.
3. Calculation: $8.9 \times 10^3 = 8900\,kg/m^3$.
Final Answer: The density is $8900\,kg/m^3$.
The standard "Metre" was originally defined in 1791 as one ten-millionth of the distance from the equator to the North Pole. Today, it is defined more precisely using the speed of light!
2.0 Measurement of Length: Least Count
Every measuring instrument has a limit to its precision. This smallest value that can be measured accurately by an instrument is called its Least Count (L.C.). A standard metre scale has a least count of $1\,mm$ (or $0.1\,cm$). To measure even smaller lengths, we use instruments like the Vernier Callipers.
The Vernier Principle
A Vernier Calliper consists of two scales: the Main Scale (fixed) and the Vernier Scale (movable). It works on the principle that $n$ divisions of the vernier scale are equal to $(n-1)$ divisions of the main scale.
Least Count (Vernier Constant): It is the difference between the value of one main scale division and one vernier scale division.
Least Count of Vernier Callipers
$$L.C. = \frac{\text{Value of 1 Main Scale Division}}{\text{Total Number of Divisions on Vernier Scale}}$$
Example: If 1 MSD = $1\,mm$ and Vernier has 10 divisions, L.C. = $0.1\,mm$ or $0.01\,cm$.
2.1 How to Read Vernier Callipers
To find the total reading of an object, we use the following formula:
Always check the instrument when jaws are closed.
1. Positive Zero Error: Vernier zero is to the right of Main Scale zero (Subtract from reading).
2. Negative Zero Error: Vernier zero is to the left of Main Scale zero (Add to reading).
While measuring the diameter of a rod, the main scale reads $2.4\,cm$ and the 5th division of the vernier scale coincides with a main scale division. If L.C. = $0.01\,cm$, find the diameter.
Solution:
1. Main Scale Reading (M.S.R.): $2.4\,cm$
2. Vernier Coincidence (n): $5$
3. Least Count (L.C.): $0.01\,cm$
4. Vernier Reading (V.S.R.): $n \times L.C. = 5 \times 0.01 = 0.05\,cm$
5. Total Reading: $M.S.R. + V.S.R. = 2.4 + 0.05 = 2.45\,cm$
Final Answer: The diameter of the rod is $2.45\,cm$.
The Vernier scale was invented by the French mathematician Pierre Vernier in 1631. It allowed for much higher precision in navigation and astronomy long before digital tools existed!
3.0 Measurement of Small Lengths: Screw Gauge
For measuring even smaller dimensions, like the thickness of a wire or a thin metal sheet, the Vernier Calliper is not precise enough. We use a Screw Gauge (or Micrometer), which works on the principle of a Screw and Nut. When a screw is rotated in a fixed nut, it moves forward or backward by a distance proportional to the rotation.
Key Definitions
- Pitch: The distance moved along the axis by the screw in one complete rotation of its head.
- Least Count (L.C.): The distance moved by the screw when the circular scale is rotated by one division.
Least Count of Screw Gauge
$$L.C. = \frac{\text{Pitch of the Screw}}{\text{Total No. of Divisions on Circular Scale}}$$
Standard Screw Gauge L.C. = $\frac{1\,mm}{100} = 0.01\,mm$ or $0.001\,cm$.
3.1 Backlash Error
In older instruments, due to wear and tear of the screw threads, rotating the head in the reverse direction might not immediately move the screw. this gap or "play" is called Backlash Error. To avoid this, always rotate the screw in one direction while taking a reading.
When the studs are in contact, the zero of the circular scale should coincide with the baseline of the main scale.
1. Positive: Circular zero is below the baseline (Subtract).
2. Negative: Circular zero is above the baseline (Add).
A screw gauge has a pitch of $1\,mm$ and $100$ divisions on its circular scale. While measuring the thickness of a glass plate, the pitch scale reads $3\,mm$ and the $47^{th}$ division of the circular scale coincides with the baseline. Find the thickness.
Solution:
1. Pitch Scale Reading (P.S.R.): $3\,mm$
2. L.C.: $1\,mm / 100 = 0.01\,mm$
3. Circular Scale Reading (C.S.R.): $47 \times 0.01 = 0.47\,mm$
4. Total Thickness: $P.S.R. + C.S.R. = 3 + 0.47 = 3.47\,mm$
Final Answer: The thickness of the glass plate is $3.47\,mm$.
The Spherometer is a relative of the screw gauge used primarily by opticians to measure the curvature of lenses and mirrors. It works on the same screw-thread principle!
4.0 Measurement of Time: The Simple Pendulum
Time is a fundamental dimension in Physics. A Simple Pendulum is an ideal arrangement consisting of a heavy point mass (called a bob) suspended by a weightless, inextensible string from a rigid support. It works on the principle of periodic motion.
Pendulum Terminology
- Oscillation: One complete to-and-fro motion of the bob (e.g., from mean position to one extreme, to the other extreme, and back to mean).
- Effective Length ($l$): The distance from the point of suspension to the centre of gravity of the bob.
- Time Period ($T$): The time taken to complete one full oscillation.
Time Period of a Simple Pendulum
$$T = 2\pi \sqrt{\frac{l}{g}}$$
Where: $l$ = effective length, $g$ = acceleration due to gravity ($9.8\,m/s^2$).
4.1 Factors Affecting Time Period
Based on the formula, we can observe the Laws of Simple Pendulum:
- Law of Length: The time period is directly proportional to the square root of its effective length ($T \propto \sqrt{l}$).
- Law of Gravity: The time period is inversely proportional to the square root of acceleration due to gravity ($T \propto 1/\sqrt{g}$).
- Independence of Mass: The time period does not depend on the mass or material of the bob.
- Independence of Amplitude: For small oscillations, $T$ is independent of the amplitude.
A Seconds Pendulum is a special pendulum whose time period is exactly 2 seconds. This means it takes 1 second to go from one extreme to the other.
Its effective length on Earth is approximately 1 metre ($99.4\,cm$).
How much will the time period of a pendulum change if its effective length is made four times its original length?
Solution:
1. Original Time Period: $T_1 = 2\pi \sqrt{l_1/g}$
2. New length: $l_2 = 4l_1$
3. New Time Period: $T_2 = 2\pi \sqrt{4l_1/g}$
4. $T_2 = 2 \times (2\pi \sqrt{l_1/g}) = 2T_1$
Final Answer: The time period will be doubled.
Pendulum clocks go slower in summer and faster in winter. Why? In summer, the metal rod expands (thermal expansion), increasing $l$, which increases $T$. In winter, the rod contracts, $l$ decreases, and the clock gains time!
5.0 Significant Figures and Errors
In experimental physics, no measurement is ever "perfect." The precision of a measurement is indicated by the number of Significant Figures it contains. These are the digits in a measured quantity that are known with confidence, plus one last digit that is uncertain.
Rules for Significant Figures
- All non-zero digits are significant (e.g., $125$ has three).
- Zeros between non-zero digits are significant (e.g., $1005$ has four).
- Leading zeros (to the left of the first non-zero digit) are not significant (e.g., $0.002$ has one).
- Trailing zeros in a number with a decimal point are significant (e.g., $2.50$ has three).
5.1 Types of Experimental Errors
Errors in measurement can be broadly classified into two categories based on their origin:
| Error Type | Cause | How to Reduce |
|---|---|---|
| Systematic Errors | Faulty instruments (Zero error) or poor calibration. | Use better instruments or apply mathematical corrections. |
| Random Errors | Unpredictable fluctuations in environment or observer's reflex. | Take multiple readings and calculate the mean. |
Percentage Error
$$\% \text{ Error} = \frac{\text{Experimental Value} - \text{True Value}}{\text{True Value}} \times 100$$
This is a common error while reading a scale. It occurs when the observer's eye is not vertically above the mark being read. To avoid this, always look straight down at the graduation mark.
How many significant figures are in the following: (a) 0.00502 and (b) 1.20 x 10³?
Solution:
1. For (a): The leading zeros are non-significant. '5', '0', and '2' are significant. Total = 3.
2. For (b): In scientific notation, all digits before the power of ten are significant. '1', '2', and '0' are significant. Total = 3.
Final Answer: Both (a) and (b) have 3 significant figures.
In 1999, NASA lost the Mars Climate Orbiter ($125 million) because one engineering team used Imperial units (pounds-force) while another used S.I. units (Newtons). A simple unit conversion error led to the spacecraft crashing!