Measurements and Experimentation

A. Measurement Definition:

Measurement is the process of comparison of a given physical quantity with an internationally accepted standard quantity of the same nature, called a unit.

where $n$ is the numerical value (magnitude) and $u$ is the unit. Notice that $n \propto \frac{1}{u}$ (the larger the unit, the smaller the numerical magnitude).

B. The Seven Fundamental SI Units:

A. Principle of Vernier:

A Vernier Calliper employs two scales: a stationary Main Scale (MS) graduated in millimeters ($1\text{ MSD} = 1\text{ mm} = 0.1\text{ cm}$), and a sliding Vernier Scale (VS) having $n$ divisions that coincide with $(n - 1)$ divisions of the main scale.

B. Least Count (Vernier Constant, VC):

For standard metric calipers where $1\text{ MSD} = 1\text{ mm} = 0.1\text{ cm}$ and $n = 10\text{ divisions}$:

C. Reading Formula:

where $\text{MSR}$ is Main Scale Reading and $\text{VCD}$ is Vernier Coinciding Division.

D. Zero Errors and Corrections:

• No Zero Error: The zero mark of the vernier scale aligns perfectly with the zero mark of the main scale when jaws touch.
• Positive Zero Error: The zero of the vernier lies to the right of the zero of the main scale. If the $p^{\text{th}}$ vernier division coincides: $\text{Zero Error} = +(p \times \text{LC})$.
• Negative Zero Error: The zero of the vernier lies to the left of the zero of the main scale. If the $q^{\text{th}}$ vernier division coincides: $\text{Zero Error} = -[(n - q) \times \text{LC}]$.
• Universal Correction: $\mathbf{\text{Corrected Reading} = \text{Observed Reading} - (\text{Zero Error})}$. (Subtract positive error; add negative error).

A. Principle of the Screw:

The linear distance advanced by the screw along its main axis is directly proportional to the rotation given to its circular head.

B. Pitch and Least Count:

• Pitch ($p$): The linear distance moved by the spindle on the main scale in one complete rotation ($360^\circ$) of the circular head. Usually $p = 1\text{ mm}$ or $0.5\text{ mm}$.
• Least Count (LC): $$\mathbf{\text{LC} = \frac{\text{Pitch of the Screw}}{\text{Total Number of Circular Scale Divisions (CSD)}}}$$ For a screw gauge with pitch $1\text{ mm}$ and $100$ circular divisions: $$\text{LC} = \frac{1\text{ mm}}{100} = 0.01\text{ mm} = \mathbf{0.001\text{ cm}}$$

C. Total Reading & Zero Errors:

• Positive Zero Error: The zero mark on the circular scale lies below the baseline/index line when anvil and spindle touch. $\text{Error} = +(p \times \text{LC})$.
• Negative Zero Error: The zero mark on the circular scale lies above the baseline. $\text{Error} = -[(N - q) \times \text{LC}]$, where $N$ is total circular divisions.
• Backlash Error: Loose mechanical play between the screw threads and nut caused by wear and tear. Prevented by rotating the thimble in only one direction during a measurement.

A. Definitions:

• Ideal Simple Pendulum: A heavy point mass (bob) suspended from a rigid, frictionless support by a light, inextensible, weightless string.
• Effective Length ($l$): Distance from the point of suspension ($S$) to the center of gravity ($G$) of the bob: $$\mathbf{l = L + r}$$ where $L$ is the length of the string and $r$ is the radius of the spherical bob.
• Time Period ($T$): Time taken to complete one full oscillation ($A \to B \to A \to C \to A$).
• Frequency ($f$): Number of oscillations completed per second: $f = \frac{1}{T}$ (in Hertz, $\text{Hz}$ or $\text{s}^{-1}$).

B. Galileo's Laws of the Simple Pendulum:

1. Law of Isochronism: The time period $T$ of a simple pendulum is independent of its amplitude of oscillation, provided the amplitude is small ($< 4^\circ$).
2. Law of Mass: The time period $T$ is completely independent of the mass, material, and size of the bob.
3. Law of Length: The time period is directly proportional to the square root of its effective length: $$T \propto \sqrt{l} \iff T^2 \propto l$$
4. Law of Gravity: The time period is inversely proportional to the square root of acceleration due to gravity: $$T \propto \frac{1}{\sqrt{g}}$$

C. Universal Formula & Determination of $g$:

A graph of $T^2$ against $l$ is a straight line passing through the origin. The slope of this line is:

Seconds' Pendulum: A pendulum with a time period of exactly $T = 2.0\text{ seconds}$ (takes $1\text{ s}$ for a half-swing). Its effective length on Earth ($g \approx 9.8\text{ m/s}^2$) is approximately $\mathbf{l \approx 1.0\text{ meter}} = 99.4\text{ cm}$.

Problem 1: In a vernier callipers, $19$ divisions of the main scale coincide with $20$ divisions of the vernier scale. If $1\text{ MSD} = 1\text{ mm}$, find its least count. When measuring the diameter of a cylinder, the main scale reads $2.8\text{ cm}$ and the $14^{\text{th}}$ vernier division coincides with a main scale mark. Find the diameter of the cylinder.

Solution:

1. Least Count:

2. Observed Diameter:

Problem 2: A simple pendulum of effective length $1.0\text{ m}$ completes $20$ oscillations in $40.4\text{ seconds}$. Calculate the acceleration due to gravity $g$ at that location (take $\pi = 3.1416$).

Solution:

1. Time period $T$ for one oscillation:

2. Using $T^2 = \frac{4\pi^2 l}{g}$: