Newtons's law of Gravitiation and gravitational fiels

Newton's Law of Gravitation and Gravitational Fields

Learning Objectives

  • 4.1 Newton's Law of Universal Gravitation (L1)

    • All objects with mass attract one another with a gravitational force.

    • Magnitude of this force calculated using:
      F_g = G rac{m_1 m_2}{r^2}

  • 4.2 Gravitational Fields (L2)

    • Objects with mass produce a gravitational field in the space surrounding them.

    • Gravitational force on an object arises from the presence of a gravitational field:
      F_{ ext{weight}} = mg

  • 4.3 Work in Gravitational Fields (L3)

    • Work is done when a mass moves in a gravitational field, changing its potential energy. Relevant equations:

    • Change in gravitational potential energy:
      riangle E_p = mg riangle h

    • Work done:
      W = F_s
      W = riangle E

    • Kinetic energy:
      E_k = rac{1}{2}mv^2

  • Gravitational Field Strength: Defined as the net force per unit mass at a point in the field. Relationship:
    g = rac{F_g}{m} = rac{GM}{r^2}

Success Criteria

  • By the end of the week, students should be able to:

    • Apply Newton's Law of Universal Gravitation to calculate gravitational forces.

    • Calculate gravitational field strength and its relationship to weight (
      F = mg).

    • Calculate changes in gravitational potential energy using:

    • riangle E_p = mg riangle h (near Earth's surface)

    • E_p = - rac{GMm}{r} (in space)

    • Understand that gravitational fields surround all masses, causing forces.

Homework - Due 2nd Lesson

  • Review sections 4.1, 4.2, 4.3:

    • Make notes

    • Answer key questions Chapter 4 review (Q1 to 18)

    • Complete STAWA Problem Set 5 (Q1-Q11)

Historical Context: Newton's Law of Universal Gravitation

  • Historical Development

    • 1687 - In Principia Mathematica:

    • Introduced calculus and outlined laws of motion.

    • First explanation of planetary motion.

    • Early inspiration from the falling apple linked to the Moon's orbit.

  • Contributors: Sir Isaac Newton, Gottfried Wilhelm Leibniz, Tycho Brahe, Johannes Kepler.

  • Cavendish Experiment

    • Henry Cavendish accurately measured the gravitational constant, G, in 1798 using a torsion balance to observe the attraction between lead spheres at a known distance. This measurement allowed the determination of G from observed forces.

Worked Examples

Example 4.1.1: Gravitational Attraction Between Small Objects

  • Scenario: 90.0 kg person and 75.0 kg friend separated by 80.0 cm.

  • Calculate gravitational force:

    • Using the equation:
      F_g = G rac{m_1 m_2}{r^2}

Example 4.1.2: Gravitational Attraction Between Massive Objects

  • Scenario: Sun and Earth.

  • Given:

    • m_{sun} = 1.99 imes 10^{30} ext{ kg}

    • m_{Earth} = 5.97 imes 10^{24} ext{ kg}

    • Distance between Sun and Earth:
      r = 1.50 imes 10^{11} ext{ m}

Example 4.1.3: Acceleration Caused by a Gravitational Force

  • Scenario: Gravitational force between Moon and Earth, with:

    • m_{Moon} = 7.35 imes 10^{22} ext{ kg}

    • F_{g} = 1.98 imes 10^{20} ext{ N}

  • Calculate the acceleration of both bodies caused by the force:

    • a_{Moon} = rac{F_{g}}{m_{Moon}}

    • a_{Earth} = rac{F_{g}}{m_{Earth}}

Example 4.1.5: Calculating Apparent Weight in a Lift

  • Scenario: A 79.0 kg student in an upward-accelerating lift (1.26 m/s²):

    • Apparent weight calculation during acceleration (
      F_a = m(g + a)) and at constant speed (no acceleration).

  • Repeat calculations for varying accelerations and directions:

    • Upward: F_a = m(g + a)

    • Constant speed: F_a = mg

    • Downward: F_a = m(g - a)

Summary

  • Gravitational Attraction

    • All objects with mass attract each other, quantified by:
      F = G rac{m_1 m_2}{r^2}

  • The weight of an object is the force due to gravity acting on it, defined as:
    F_{ ext{weight}} = mg

  • Acceleration due to gravity near Earth's surface:

    • g = G rac{M_{Earth}}{r^2} = 9.80 ext{ m/s}^2

  • Apparent Weight can be affected by vertical acceleration.

  • Implications of Weightlessness: Objects in free-fall (like astronauts in orbit) experience weightlessness due to constant acceleration towards Earth while maintaining orbital velocity.

Homework Questions

  1. Discuss proportionalities in Newton's law of universal gravitation.

  2. Explain the symbol
    r in the context of the gravitational equation.

  3. Calculate gravitational forces between the Sun and Mars, along with other stellar influences.

  4. Understand the concept of apparent weight vs. weight during dynamic movement (e.g., lifts).