Introduction to Physical Quantities, Measurement, and Density

Physical Quantities and SI Units

• Definition of Physical Quantity: A physical quantity is a quantity which can be measured. It is always measured of natural non-living objects.

• Components of a Physical Quantity: Every physical quantity consists of a numerical magnitude and a unit.     * Example: If the length of a student is $104\,\text{cm}$, then $104$ is the numerical magnitude and $\text{cm}$ is the unit of measurement.

• Basic SI Units (Base Units):     * Length: The base unit is the meter, represented by the symbol $\text{m}$.     * Mass: The base unit is the kilogram, represented by the symbol $\text{kg}$.     * Time: The base unit is the second, represented by the symbol $\text{s}$.     * Electric Current: The base unit is the ampere, represented by the symbol $\text{A}$.     * Temperature: The base unit is the Kelvin, represented by the symbol $\text{K}$.     * Light Intensity: The base unit is the candela, represented by the symbol $\text{cd}$.     * Amount of Substance: The base unit is the mole, represented by the symbol $\text{mol}$.

Multiplication Factors and Prefixes

• Large Scale Prefixes:     * Tera (T): Multiplication factor of $10^{12}$ ($1,000,000,000,000$).     * Giga (G): Multiplication factor of $10^9$ ($1,000,000,000$).     * Mega (M): Multiplication factor of $10^6$ ($1,000,000$).     * Kilo (k): Multiplication factor of $10^3$ ($1,000$).

• Small Scale Prefixes:     * Deci (d): Multiplication factor of $10^{-1}$, representing $0.1$ or $\frac{1}{10}$.     * Centi (c): Multiplication factor of $10^{-2}$, representing $0.01$ or $\frac{1}{100}$.     * Milli (m): Multiplication factor of $10^{-3}$, representing $0.001$ or $\frac{1}{1000}$.     * Micro (\mu): Multiplication factor of $10^{-6}$, representing $0.000001$ or $\frac{1}{1,000,000}$.     * Nano (n): Multiplication factor of $10^{-9}$, representing $0.000000001$ or $\frac{1}{1,000,000,000}$.     * Pico (p): Multiplication factor of $10^{-12}$, representing $0.000000000001$ or $\frac{1}{1,000,000,000,000}$.     * Femto (f): Multiplication factor of $10^{-15}$, representing $0.000000000000001$ or $\frac{1}{1,000,000,000,000,000}$.

Significant Figures

• Rules for Counting Significant Digits:     * Non-zero Digits: All non-zero digits are significant.     * In-between Zeros: Any zeros contained between non-zero digits count as significant. For example, $300042$ has $6$ significant digits.     * Leading Zeros: Leading zeros never count as significant figures. For example, $0.000034$ has $2$ significant digits.     * Trailing Zeros:         * If there is a decimal point, trailing zeros count. For example, $0.0002500$ has $4$ significant digits.         * If there is no decimal point, trailing zeros may or may not count. The conservative approach is to assume they are not significant. For example, $190000$ is treated as having $2$ significant digits (though it could theoretically have up to $6$).     * General Example: In the value L > 0.0050830, there are leading zeros (not significant), in-between zeros (significant), non-zero digits (significant), and trailing zeros with a decimal (significant).

• Calculations with Significant Digits:     * Addition and Subtraction: Round the final answer to the least number of decimal places found in the initial data values.         * Example 1: $1.457 + 83.2 = 84.657$, which rounds to $84.7$ (matching the one decimal place in $83.2$).         * Example 2: $0.0367 - 0.004322 = 0.032378$, which rounds to $0.0324$ (matching the four decimal places in $0.0367$).     * Multiplication and Division: Round the final answer to the least number of significant digits found in the initial data values.         * Example 1: $4.36 \times 0.00013 = 0.0005668$, which rounds to $0.00057$ (matching the two significant digits in $0.00013$).         * Example 2: $\frac{12.300}{0.0230} = 534.78261$, which rounds to $535$ (matching the three significant digits in $0.0230$).

Accuracy and Errors

• Types of Errors:     * Personal Errors: Caused by carelessness. These should never be discussed in the conclusion; if they occur, the experiment should simply be redone.     * Systematic Errors: These represent reproducible inaccuracies caused by an instrument or another factor that can be corrected. In a systematic error, all errors are in the same direction. If detected, the experiment should be redone.     * Random Errors: Unpredictable and unknown variations that cannot be eliminated. Statistical analysis is used to communicate and account for this type of error.

Measurement Techniques: Simple Pendulum

• Time Period (T): The time it takes for one complete oscillation to occur.

• Measurement Strategy: To reduce the impact of human reaction time errors, it is preferable to measure the time for multiple oscillations (e.g., $10$ or $20$) and then calculate the average time per oscillation.

• Step-by-Step Procedure:     1. Set up the pendulum: Suspend it freely so the bob is at rest.     2. Displace the bob: Gently move the bob to one side and release it.     3. Start the timer: Use a stopwatch and start it as the bob passes a reference point, such as the vertical center.     4. Stop the timer: Stop the stopwatch after the bob completes the desired count of oscillations.     5. Calculate the period: Divide total time by the number of oscillations ($T = \frac{\text{total time}}{\text{number of oscillations}}$).     * Example: Timing $10$ oscillations results in $20\,\text{seconds}$. The period is $\frac{20\,\text{seconds}}{10\,\text{oscillations}} = 2\,\text{seconds}$ per oscillation.

• Improving Accuracy: Use a stopwatch with high resolution and ensure the pendulum swings in a small angle.

• Fiducial Mark: Utilizing a fiducial mark (a reference point) helps ensure more accurate timing during the start and stop of the stopwatch.

Comparison of Length Measuring Tools

• Vernier Caliper: Used for both external and internal measurements. It has a least count of $0.01\,\text{mm}$.

• Micrometer Screw Gauge: Used specifically for external measurements. It has a higher precision than the vernier caliper, with a least count of $0.001\,\text{mm}$.

Scalars and Vectors

• Scalar Quantity: Possesses only magnitude and no direction.     * Examples: Length, Area, Volume, Speed, Mass, Density, Pressure, Temperature, Energy, Entropy, Work, Power, Time, Distance.

• Vector Quantity: Possesses both magnitude and direction.     * Examples: Displacement, Velocity, Acceleration, Momentum, Force, Lift, Drag, Thrust, Weight, Gravity, Impulse.

Mass vs. Weight

• Mass:     * A measure of the amount of matter in an object.     * Remains constant regardless of the object's location in the universe.     * Measured using a balance.     * Units: $\text{kg}$, $\text{g}$, $\text{mg}$.

• Weight:     * A measure of the gravitational force acting on an object.     * Varies depending on the object's location relative to Earth or other large bodies in space.     * Measured using a spring scale.     * Units: Newtons ($\text{N}$).

Density

• Definition: Density ($\rho$) is a measure of how much matter occupies a given amount of space. This symbol $\rho$ is called "rho".

• Formula: $\rho = \frac{m}{V}$, where $m$ is mass and $V$ is volume.

• States of Matter: Density is a useful measure for all states of matter (solids, liquids, and gases).

• Unit of Density: $\text{kg/m}^3$.

• Floating and Sinking Principles:     * Floating: Occurs if the density of the object is less than the density of water (\rho_{\text{object}} < \rho_{\text{water}}).     * Neutral Buoyancy: Occurs if the density of the object equals the density of water ($\rho_{\text{object}} = \rho_{\text{water}}$).     * Sinking: Occurs if the density of the object is greater than the density of water (\rho_{\text{object}} > \rho_{\text{water}}).

Pressure

• Definition: Pressure ($p$ or $P$) is the force applied perpendicular to the surface of an object per unit area over which that force is distributed.

• The Unit of Pressure (Pascal): One pascal ($1\,\text{Pa}$) is defined as a force of one newton ($1\,\text{N}$) applied to an area of one square meter ($1\,\text{m}^2$).     * $1\,\text{Pa} = 1\,\text{N/m}^2$

• Pressure in Liquids:     * Calculated by the formula: $P = h\rho g$     * Where $h$ is the depth of water, $\rho$ is the density of water, and $g$ is the gravitational field strength.

• Total Pressure: The sum of the liquid pressure and the atmospheric pressure.     * $P_{\text{Total}} = \rho gh + P_{\text{atm}}$

• Atmospheric Pressure: The pressure exerted by the weight of the atmosphere. At sea level, it has a mean value of $101,325\,\text{Pascals}$.

Questions & Discussion

Q: A student measures the depth of water dripping into a jar every minute. What equipment does she use?

• A: 1. A ruler (to measure depth). 2. A stopwatch or clock (to measure the one-minute intervals).

Q: Determine the average depth of 2.5 mm in meters.

• Calculation: Since $1000\,\text{mm} = 1\,\text{m}$, then $\frac{2.5}{1000} = 0.0025\,\text{m}$ or $2.5 \times 10^{-3}\,\text{m}$.

Q: State what is meant by the term weight.

• A: Weight is the force or pull of gravity acting on an object.

Q: Equipment on a distant planet has a mass of 350 kg. Gravity is 7.5 m/s². What is its weight?

• Calculation: $Weight = Mass \times Acceleration\,due\,to\,gravity \rightarrow 350\,\text{kg} \times 7.5\,\text{m/s}^2 = 2600\,\text{N}$.

Q: An inflated balloon has a mass of 80 g and a volume of 0.30 m³. The planet's atmosphere density is 0.35 kg/m³. Predict the balloon's movement.

• Calculation:     * Convert mass to kg: $80\,\text{g} = 0.08\,\text{kg}$.     * Calculate density: $\rho = \frac{m}{V} = \frac{0.08}{0.30} = 0.27\,\text{kg/m}^3$.

• Prediction: The balloon moves/floats up because its density ($0.27\,\text{kg/m}^3$) is less than the density of the atmosphere ($0.35\,\text{kg/m}^3$).

Q: Explain why a crawler-board prevents men from falling through a weak roof.

• A: Use of the board provides a larger area in contact with the roof. This spreads the weight and force out, resulting in a lower pressure on the roof.

Q: Calculate the pressure of a 400 N crawler-board holding two men with a total weight of 1600 N on an area of 0.8 m².

• Calculation:     * Total Force ($F$) = $400\,\text{N} + 1600\,\text{N} = 2000\,\text{N}$.     * Pressure ($P$) = $\frac{F}{A} = \frac{2000\,\text{N}}{0.8\,\text{m}^2} = 2500\,\text{Pa}$ (or $\text{N/m}^2$).

Q: Calculate the pressure under a girl's feet if her mass is 50 kg and the area of her shoes is i) 2 cm² and ii) 200 cm².

• Note: This specific calculation was provided as a prompt in the slides for student practice, emphasizing the inverse relationship between area and pressure.