OSUN STATE UNIVERSITY GENERAL PHYSICS (PHY 101) NOTES

OSUN STATE UNIVERSITY GENERAL PHYSICS (PHY 101) NOTES

Course Overview

  • Course Code: PHY 101
  • Unit: 3
  • Lecturer: I.A. Akanni (PhD)
  • Semester: Harmattan Semester, 2024/2025
  • Department: Physics, Faculty of Basic and Applied Sciences

Course Contents

  • Newton’s Law of Gravitation
  • Kepler’s Laws of Planetary Motion
  • Gravitational Potential Energy
  • Escape Velocity
  • Satellite Motion and Orbits

Newton’s Law of Gravitation

Universal Law of Gravitation
  • Definition: States that every particle in the universe attracts every other particle with a force that is:
    • Directly proportional to the product of their masses
    • Inversely proportional to the square of the distance between their centers.
  • Formula: (F=GM<em>1M</em>2r2)(F = G \frac{M<em>1 M</em>2}{r^2}) Where:
    • $F$ = gravitational force between two masses
    • $M_1$ = mass of the first body (e.g., Earth)
    • $M_2$ = mass of the second body (e.g., Moon)
    • $r$ = distance between the centers of the two bodies
    • $G$ = gravitational constant ($6.67 \times 10^{-11} \text{ N m}^{2}/\text{kg}^{2}$)
Example Calculations

  1. Example 1: Calculate the gravitational force between Earth and the Moon.

    • Given:
      • $M_{Earth} = 6.0 \times 10^{24} \text{ kg}$
      • $M_{Moon} = 7.4 \times 10^{22} \text{ kg}$
      • $r = 345,000 \text{ km} = 3.45 \times 10^{8} \text{ m}$
    • Calculation:
      • F=G(6.0×1024)(7.4×1022)(3.45×108)2F = G \frac{(6.0 \times 10^{24})(7.4 \times 10^{22})}{(3.45 \times 10^{8})^2}
      • Result: F≈2.49×1020 NF \approx 2.49 \times 10^{20} \text{ N}

  2. Example 2: Calculate the gravitational force of attraction between Earth and a 70 kg man standing at sea level.

    • Given:
      • $M_{Earth} = 5.98 \times 10^{24} \text{ kg}$
      • Distance from the Earth’s center, $r = 6.38 \times 10^{6} \text{ m}$
    • Calculation:
      • F=G(5.98×1024)(70)(6.38×106)2F = G \frac{(5.98 \times 10^{24})(70)}{(6.38 \times 10^{6})^2}
      • Result: F≈685 NF \approx 685 \text{ N}
Gravitational Field
  • Definition: A gravitational field is a region where a mass experiences a force due to gravitational attraction.
    • Characteristics:
    • Direction is always towards the center of the mass causing the field.
    • Gravitational forces are always attractive.
    • Has infinite range, affecting all objects in the universe.
    • The strength of the field varies with mass:
      • Greater gravitational force around large masses (planets)
      • Negligible force around small masses (atoms)
Gravitational Field Strength
  • Definition: The gravitational field strength (or gravitational acceleration) at a point is defined as the force per unit mass experienced by a test mass.
  • Formula: g=Fmg = \frac{F}{m} Where:
    • $g$ = gravitational field strength ($\text{m} s^{-2}$)
    • $F$ = force due to gravity (weight, in Newtons)
    • $m$ = mass of the test mass (in kg)
Interpretation of Mass and Weight

  • Mass:

    • Definition: The amount of matter in an object, an intrinsic property that does not change regardless of location.
    • Units: kg, g, etc.
    • Characteristics: Scalar quantity with only magnitude; constant throughout the universe.
    • Mass influences inertia: Higher mass = greater resistance to changes in motion.
    • Example: On Jupiter, humans cannot stand fully due to increased gravitational force leading to increased weight.

  • Weight:

    • Definition: The force exerted on an object due to gravity, dependent on both the object's mass and the strength of the gravitational field.
    • Units: Newtons (N).
    • Characteristics: Vector quantity with both magnitude and direction; varies by location.
    • Example Calculation (Weight of a 10 kg stone):
    • On Earth: W=mg=10kg×9.81m/s2=98.1NW = mg = 10 kg \times 9.81 m/s^2 = 98.1 N
    • On Moon (gravitational acceleration = 1.6 m/s^2): W=10kg×1.6m/s2=16NW = 10 kg \times 1.6 m/s^2 = 16 N

Kepler’s Law of Planetary Motion

  • Introduction:
    • Describes the motion of planets around the Sun, based on laws formulated by Johannes Kepler from observations made by Tycho Brahe.
Three Laws of Planetary Motion

  1. Law of Orbits (First Law):

    • All planets orbit the Sun in elliptical paths, with the Sun at one of the foci of the ellipse.
    • Ellipse Definition: An elongated circle with two focal points, where distance from the center varies, affecting planet speed during orbit.
    • Formula for ellipse:
      • (x2a2)+(y2b2)=1(\frac{x^2}{a^2}) + (\frac{y^2}{b^2}) = 1 (where $a$ = semi-major axis, $b$ = semi-minor axis)

  2. Law of Equal Areas (Second Law):

    • A line (radius vector) from the Sun to a planet sweeps out equal areas in equal times, meaning a planet speeds up as it approaches the Sun (perihelion) and slows down as it moves away (aphelion).
    • Conservation of angular momentum ensures the radius vector covers constant area over constant time intervals.
    • Mathematically expressed as:
      • dAdt=constant\frac{dA}{dt} = constant (where $A$ is the area swept out)

  3. Harmonic Law (Third Law):

    • The square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit, expressed as:
      • (T2∝a3)(T^2 \propto a^3)
        Where:
      • $T$ = orbital period (in years)
      • $a$ = semi-major axis (in astronomical units)
    • For any two planets: T<em>12T</em>22=a<em>13a</em>23\frac{T<em>1^2}{T</em>2^2} = \frac{a<em>1^3}{a</em>2^3}

Gravitational Potential Energy (U or PEG)

  • Definition: The energy stored in an object due to its position in a gravitational field.
  • Factors:
    • Dependence on mass ($m$), height ($h$) above a reference point, and gravitational field strength ($g$).
  • Formula (near Earth's surface): U=mghU = mgh
    • Interpretation: Higher objects with greater mass have more gravitational potential energy, indicating increased capacity for doing work when falling.
Gravitational Potential Formula
  • For objects under non-uniform gravitational fields at distance $r$ from a mass $M$: U=−GMmrU = -\frac{GMm}{r}
    • Significance of Negative Value: Indicates potential energy that decreases as the object moves closer to the massive body, consistent with gravitational attraction.
Gravitational Potential (V)
  • Definition: Work needed to move a unit mass from infinity to a point in a gravitational field without acceleration.
  • Formula: V=−GMrV = -\frac{GM}{r}
    • Positive to negative potential transition reflects gravitational energy decrease as objects approach mass.
Examples of Gravitational Potential Energy Calculations
  1. Example 1: Calculate gravitational potential at a distance of 10,000 km from Earth's center.
    • V=−6.674×10−11×5.97×102410,000,000≈−39.8 million J/kgV = -\frac{6.674 \times 10^{-11} \times 5.97 \times 10^{24}}{10,000,000} \approx -39.8 \text{ million J/kg}
  2. Example 2: Calculate gravitational potential energy of a satellite weighing 500 kg at 10,000 km.
    • U=500imes−39.8×106=−19.9 billion JU = 500 imes -39.8 \times 10^6 = -19.9 \text{ billion J}
Work Calculation
  • Work needed to raise an object from Earth's surface to height $h$: W=U<em>final−U</em>initialW = U<em>{final} - U</em>{initial}
    • Based on differences in potential energy at the initial and final positions.

Escape Velocity

  • Definition: Minimum speed required for an object to escape gravitational pull without further propulsion.
  • Key Principle: Total energy at the surface must be equal to that at infinity to escape gravitational influence.
    • Key Equations:
    • Conservation of energy at the planet's surface:
      12mv2−GMmR=0\frac{1}{2}mv^2 - \frac{GMm}{R} = 0
    • Results in escape velocity formula:
      v=2GMRv = \sqrt{\frac{2GM}{R}}
    • Where:
    • $G$ = gravitational constant ($6.674 \times 10^{-11} \text{ N m}^{2}/\text{kg}^{2}$)
    • $M$ = mass of the celestial body
    • $R$ = radius of the celestial body
Escape Velocity Calculations
  • Example 1: Estimate Earth's escape velocity.
    • v=2×6.674×10−11×5.97×10246.37×106≈11.2extkm/sv = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 5.97 \times 10^{24}}{6.37 \times 10^6}} \approx 11.2 ext{ km/s}
  • Example 2: Escape velocity at twice Earth's radius.
    • v=GM2R≈7.9km/sv = \sqrt{\frac{GM}{2R}} \approx 7.9 km/s

Satellite Motion and Orbits

  • Definition of Satellites: Objects in orbit around a celestial body; the path of this orbit is called the orbit.
Types of Orbits
  1. Circular Orbit: Satellite distance from the center remains constant, speed constant balanced by gravitational pull.
  2. Elliptical Orbit: Satellite varies distance to body; moves closer at perihelion and farther at aphelion.
  3. Geostationary Orbit: Satellite appears stationary with respect to surface clocking with the Earth’s rotation.
  4. Polar Orbit: Satellite moves over poles, allowing observation of Earth surface over time.
Maintaining Orbit
  • Gravitational force provides centripetal force that maintains orbits.
  • Key Relationships:
    • Gravitational force balancing centripetal force:
    • F<em>g=F</em>cF<em>g = F</em>c
    • Gravitational force formula:
      F=GMmr2F = G\frac{Mm}{r^2}
    • Centripetal force formula:
      Fc=mv2/rF_c = m v^2 / r
    • Resulting in orbital velocity formula:
      v=GMrv = \sqrt{\frac{GM}{r}}
    • Orbital velocity increases with decreasing radius because gravitational force increases.