Module 1 Space and Time Physics Study Notes

Introduction to Space, Time, and Physics

• Conceptual Definitions of Space and Time:

  • Space: Within the context of physics, space is the region in which objects exist and events occur. It provides the position or location of an object.

  • Time: Time represents the continuous progression of events from the past through the present and into the future. It identifies when events happen.

• The Importance of Space and Time in Physics:

  1. They provide the framework to describe motion.

  2. They are fundamental components used in measurements.

  3. They allow for the study of events and changes occurring in the universe.

• Standard Units of Measurement:

  • The standard unit of space (specifically length) is the meter ($m$).

  • The standard unit of time is the second ($s$).

• Defining Physics:

  • Physics is the branch of science dealing with the study of nature and the properties of matter and energy.

  • The subject matter encompasses heat, light, sound, electricity, magnetism, and the structure of atoms.

  • Physics relies on the scientific method, which involves the design of laws followed by verification through experiments.

  • Due to the attempt to measure quantities with the best possible accuracy, physics is also defined as the science of measurement.

• Applied Physics:

  • Applied physics is the application of physical principles to help human beings and solve practical problems.

  • It serves as a bridge or connection between the fields of Physics and Engineering.

Physical Quantities, Units, and Measurement

• Physical Quantities:

  • All quantities that can be expressed in terms of the laws of physics and can be measured are called physical quantities.

  • Examples: Distance, speed, mass, force, etc.

• Measurement:

  • Measurement is the process of comparing an unknown physical quantity with a known fixed physical quantity.

  • Magnitude expression and comparison in daily life are achieved through measurement.

• Units:

  • A unit is defined as the known fixed physical quantity used as a standard for measurement.

  • Example: Saying a classroom is $8\,\text{meters}$ long means comparing the classroom length with the standard quantity of length ($1\,\text{meter}$).

• Mathematical Expression of Physical Quantities:

  • $Q = n \times u$

  • $Q$: Physical Quantity.

  • $n$: Numerical value.

  • $u$: Standard unit.

  • Example: Mass of a stool = $15\,\text{kg}$. Here, Mass is the physical quantity, $15$ is the numerical value, and kg is the standard unit. This means the stool's mass is $15$ times the known quantity of $1\,\text{kg}$.

• Characteristics of a Standard Unit:

  1. Well-defined: Its concept must be clear.

  2. Invariance: It should not change with physical conditions such as temperature, pressure, or stress.

  3. Suitable Size: It should be neither too large nor too small.

  4. Stability: It should not change with place or time.

  5. Reproducible: It must be capable of being reproduced.

  6. International Acceptance: It must be accepted worldwide.

Classification of Units and Systems of Units

• Fundamental Quantities and Units:

  • These are quantities independent of other physical quantities.

  • In mechanics, the fundamental quantities are mass, length, and time.

  • Units associated with these are called fundamental units.

    • Mass: Kilogram ($kg$), Gram, Pound.

    • Length: Meter ($m$), Centimeter, Foot.

    • Time: Second ($s$).

• Derived Quantities and Units:

  • These are quantities derived from fundamental quantities.

  • Examples:

    • $\text{Area} = \text{Length} \times \text{Breadth} = \text{Length} \times \text{Length} = (\text{Length})^2$

    • $\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{\text{Length}}{\text{Time}}$

  • The units for these quantities are termed derived units.

• Systems of Units:

  1. F.P.S System: Length is measured in feet, mass in pounds, and time in seconds.

  2. C.G.S System: Length is measured in centimeters, mass in grams, and time in seconds.

  3. M.K.S System: Length is measured in meters, mass in kilograms, and time in seconds.

  4. S.I. System (International System of Units): An improved and extended version of the M.K.S system, modified to include more fundamental and supplementary units as science progressed to include electricity and heat.

• Fundamental S.I. Units:

  1. Length: Meter ($m$)

  2. Mass: Kilogram ($kg$)

  3. Time: Second ($s$)

  4. Temperature: Kelvin ($K$)

  5. Electric Current: Ampere ($A$)

  6. Luminous Intensity: Candela ($cd$)

  7. Quantity of Matter: Mole ($mol$)

• Supplementary S.I. Units:

  1. Plane Angle: Radian ($rad$)

  2. Solid Angle: Steradian ($sr$)

• Advantages of the S.I. System:

  1. Coherent: Derived units are easily obtained by multiplication or division of fundamental units.

  2. Rational: Uses only one unit for one physical quantity (e.g., Joule ($J$) is the unit for all forms of energy including heat, light, and mechanical).

  3. Metric: Multiples and submultiples are expressed in powers of $10$.

Dimensions and Dimensional Formulae

• Definition of Dimensions:

  • The powers to which fundamental units of mass ($M$), length ($L$), and time ($T$) are raised represent the nature of the quantity, not its magnitude.

  • Example: Area = $\text{Length} \times \text{Breadth} = [L^1] \times [L^1] = [L^2] = [M^0 L^2 T^0]$. The dimensions of area are $0$ in mass, $2$ in length, and $0$ in time.

• Dimensional Formula:

  • An expression indicating how a physical quantity depends on fundamental units.

  • Example: $\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{[L^1]}{[T^1]} = [M^0 L^1 T^{-1}]$ (Shows speed depends on length and time, but not mass).

• Dimensional Equation:

  • An equation formed by equating a physical quantity with its dimensional formula.

  • Example: $\text{Density} = [M^1 L^{-3} T^0]$.

• Table of Dimensional Formulae and Units:

  1. Force: $\text{Mass} \times \text{acceleration} = [M^1 L^1 T^{-2}]$; Unit: Newton ($N$)

  2. Work: $\text{Force} \times \text{distance} = [M^1 L^2 T^{-2}]$; Unit: Joule ($J$)

  3. Power: $\frac{\text{Work}}{\text{time}} = [M^1 L^2 T^{-3}]$; Unit: Watt ($W$)

  4. Energy (all forms): Stored work = $[M^1 L^2 T^{-2}]$; Unit: Joule ($J$)

  5. Pressure/Stress: $\frac{\text{Force}}{\text{area}} = [M^1 L^{-1} T^{-2}]$; Unit: $N\,m^{-2}$

  6. Momentum: $\text{Mass} \times \text{velocity} = [M^1 L^1 T^{-1}]$; Unit: $kg\,m\,s^{-1}$

  7. Moment of Force: $\text{Force} \times \text{distance} = [M^1 L^2 T^{-2}]$; Unit: $N\,m$

  8. Impulse: $\text{Force} \times \text{time} = [M^1 L^1 T^{-1}]$; Unit: $Ns$

  9. Strain: $\frac{\text{Change in dimension}}{\text{Original dimension}} = [M^0 L^0 T^0]$; No unit

  10. Modulus of Elasticity: $\frac{\text{Stress}}{\text{Strain}} = [M^1 L^{-1} T^{-2}]$; Unit: $N\,m^{-2}$

  11. Surface Energy: $\frac{\text{Energy}}{\text{Area}} = [M^1 L^0 T^{-2}]$; Unit: $Joule/m^2$

  12. Surface Tension: $\frac{\text{Force}}{\text{Length}} = [M^1 L^0 T^{-2}]$; Unit: $N/m$

  13. Co-efficient of Viscosity: $\frac{\text{Force} \times \text{Distance}}{\text{Area} \times \text{Velocity}} = [M^1 L^{-1} T^{-1}]$; Unit: $N/m^2$

  14. Moment of Inertia: $\text{Mass} \times (\text{radius of gyration})^2 = [M^1 L^2 T^0]$; Unit: $kg\,m^2$

  15. Angular Velocity: $\frac{\text{Angle}}{\text{time}} = [M^0 L^0 T^{-1}]$; Unit: $rad/s$

  16. Frequency: $\frac{1}{\text{Time period}} = [M^0 L^0 T^{-1}]$; Unit: Hertz ($Hz$)

  17. Area: $\text{Length} \times \text{Breadth} = [M^0 L^2 T^0]$; Unit: $m^2$

  18. Volume: $\text{Length} \times \text{breadth} \times \text{height} = [M^0 L^3 T^0]$; Unit: $m^3$

  19. Density: $\frac{\text{Mass}}{\text{volume}} = [M^1 L^{-3} T^0]$; Unit: $kg/m^3$

  20. Speed/Velocity: $\frac{\text{Distance}}{\text{time}} = [M^0 L^1 T^{-1}]$; Unit: $m/s$

  21. Acceleration: $\frac{\text{Velocity}}{\text{time}} = [M^0 L^1 T^{-2}]$; Unit: $m/s^2$

Classification Based on Dimensional Analysis

1. Dimensional Constant: Quantities with dimensions and a fixed value. Examples: Planck’s constant, gas constant, universal gravitational constant.

2. Dimensional Variable: Quantities with dimensions but without a fixed value. Examples: velocity, acceleration, force.

3. Dimensionless Constant: Quantities without dimensions but with a fixed value. Examples: $e$, $\pi$, pure numbers like $1, 2, 3$.

4. Dimensionless Variable: Quantities without dimensions and without a fixed value. Examples: angle, strain, specific gravity.

Scalar and Vector Quantities

• Scalar Quantities:

  • Defined as quantities having only magnitude but no direction.

  • Examples: Mass, length, density, volume, energy, temperature, electric charge, current, electric potential.

• Vector Quantities:

  • Defined as quantities having both magnitude and direction.

  • Examples: Displacement, velocity, acceleration, force, electric intensity, magnetic intensity.

• Vector Addition Principles:

  • Unlike scalars ($30+50=80$), vector addition accounts for direction.

  • Same Direction (Parallel): Resultant $R = F_1 + F_2$. Example: $70\,N + 90\,N = 160\,N$.

  • Opposite Directions (Straight line): Resultant $R = F_1 - F_2$. Example: $90\,N - 70\,N = 20\,N$.

  • Perpendicular Forces ($90^\circ$): The resultant $R$ is the diagonal of the rectangle/square. Calculated using Pythagoras theorem:

    • $R^2 = F_1^2 + F_2^2$

    • $R = \sqrt{F_1^2 + F_2^2}$

  • Addition methods include the Triangle Law and Parallelogram Law.

  • Vectors can be resolved into horizontal and vertical components.

Kinematics: Displacement, Velocity, and Acceleration

• Speed:

  • Distance moved per unit time.

  • Scalar quantity, measured in $m/s$ or $ms^{-1}$.

• Velocity:

  • Distance traveled in a specified direction per unit time (Displacement over time).

  • Vector quantity, measured in $m/s$ or $ms^{-1}$.

  • Formula: $v = \frac{s}{t}$, where $v$ is velocity, $s$ is displacement, and $t$ is time.

• Rectilinear Acceleration:

  • The rate of increase of velocity along a straight-line path in a unit of time.

  • Acceleration ($a$): Increasing rate of change of velocity.

  • Deceleration/Retardation: Decreasing rate of change of velocity (negative acceleration).

  • $\text{Acceleration (Deceleration)} = \frac{\text{Change in velocity}}{\text{Time taken}} = \frac{\text{Final velocity} - \text{Initial velocity}}{\text{Final time} - \text{Initial time}}$

Equations of Uniformly Accelerated Motion

• The Three Fundamental Equations:

  1. $v = u + at$

  2. $s = ut + \frac{1}{2}at^2$

  3. $v^2 = u^2 + 2as$

• Key Constraints/Reminders:

  1. From Rest: Initial velocity $u = 0$.

  2. Stops/Comes to Rest: Final velocity $v = 0$.

  3. Constant Velocity: Acceleration $a = 0$.

Examples and Exercises

• Dimensional Derivation Examples:

  • Density: $\frac{[M]}{[L^3]} = [M^1 L^{-3} T^0]$.

  • Power: $\frac{\text{Work}}{\text{Time}} = \frac{\text{Force} \times \text{Distance}}{\text{Time}} = \frac{[M^1 L^1 T^{-2}] \times [L]}{[T]} = [M^1 L^2 T^{-3}]$.

  • Co-efficient of Viscosity: $\frac{\text{Mass} \times \text{Acceleration} \times \text{Distance} \times \text{time}}{\text{length}^2 \times \text{Length}} = \frac{[M] \times [LT^{-2}] \times [L] \times [T]}{[L^3]} = [M^1 L^{-1} T^{-1}]$.

  • Angle: $\frac{\text{arc (length)}}{\text{radius (length)}} = \frac{[L]}{[L]} = [M^0 L^0 T^0]$ (Dimensionless).

• Comparison Examples:

  • Work ($[M^1 L^2 T^{-2}]$) and Power ($[M^1 L^2 T^{-3}]$) do NOT have the same dimensions.

  • Stress and Pressure both have the same dimension: $[M^1 L^{-1} T^{-2}]$.

  • Momentum ($[M^1 L^1 T^{-1}]$) and Impulse ($[M^1 L^1 T^{-1}]$) have the same dimensions.

• Practice Problem Contexts:

  • Jet travel distances based on speed and time (requires $s = vt$).

  • Average speed calculations for walking, cycling, and car travel.

  • Calculating magnitude of deceleration for a braking cyclist.

  • Bus acceleration when leaving a stop.

  • Sprinter velocity after initial acceleration from rest.

  • Distance covered in specific intervals (e.g., the last $3\,s$) for a car accelerating from rest.

  • Braking distance for a car avoiding an obstacle (goat) based on maximum retardation.