Lecture Notes: Newtonian Physics

Lecture Overview

• Course: ASTR 1P01

• Institution: Brock University

• Instructor: Prof. Barak Shoshany

General Themes

• Introduction to basic concepts in physics: This involves foundational principles that govern the behavior of matter and energy, crucial for understanding phenomena from the subatomic to the cosmic scale.

• Explanation of Newtonian mechanics and gravity: Focus on Isaac Newton's laws of motion and universal gravitation, which laid the groundwork for classical physics.

• Application of these concepts to understand celestial motion: How Newton's laws can explain and predict the orbits of planets, moons, and other astronomical bodies.

Key Concepts in Physics

• Mass: A fundamental property of matter, representing its resistance to acceleration (inertia). It is a scalar quantity, always positive, and expressed in kilograms (kg).

• Density: Defined as the mass per unit volume of a substance, providing insight into how compactly matter is distributed within an object. Measured in kg/m³.

• Momentum: A vector quantity describing the quantity of motion an object possesses. It is the product of an object's mass and its velocity, expressed in kg⋅m/s. Momentum is crucial for understanding collisions and interactions.

• Rate of Change: Describes how quickly a physical quantity changes over time. In kinematics, the primary rates of change are velocity (rate of change of position) and acceleration (rate of change of velocity).

Newtonian Mechanics

Newton's First Law of Motion

• Isaac Newton (1642-1727) was an English mathematician, physicist, and astronomer, renowned for establishing classical mechanics and co-inventing calculus. He held the Lucasian Professorship of Mathematics at Cambridge University.

• Published Mathematical Principles of Natural Philosophy (often referred to as Principia Mathematica) in 1687, a seminal work that introduced his three laws of motion and the law of universal gravitation, fundamentally altering scientific understanding.

• Definition: Newton's First Law, also known as the Law of Inertia, states that an object at rest remains at rest, and an object in motion continues in motion with the same speed and in the same direction unless acted upon by a net external force.

• Key points:

  • This law highlights the concept of inertia: the natural tendency of an object to resist changes in its state of motion.

  • With the introduction of his laws, the motion of celestial bodies could be understood and mathematically analyzed, facilitating the derivation of Kepler's laws from a more fundamental physical basis.

  • Applicable universally to both terrestrial objects (e.g., a book on a table) and celestial objects (e.g., a planet orbiting the Sun). Example: Dropping a ball on Earth or Mars follows the same laws, though the specific acceleration due to gravity differs.

Speed vs. Velocity

• Speed: A scalar quantity indicating how fast an object is moving, without regard to direction. It measures the magnitude of the rate of change of position (e.g., 100 km/h or 27.8 m/s).

• Velocity: A vector quantity that includes both the magnitude (speed) and the direction of an object's motion; for example, 100 km/h due north or 20 m/s at an angle of 30^{\circ} to the horizontal.

The Concept of Force

• Definition: A vector quantity defined as an interaction that, when unopposed, will change the motion of an object. More specifically, a force causes an object to accelerate.

• Measured in newtons (N), where 1\text{ N} = 1\text{ kg} \cdot \text{m/s}^2.

• Example forces:

  • A push or a pull.

  • A force of 50 N acting south.

  • A force of 100 N acting north-east.

• Generally, on Earth, forces such as friction (opposing motion between surfaces) and air resistance (drag force from the atmosphere) prevent continuous motion at constant velocity, as they act as external forces to slow objects down.

Examples of Motion in Space

• In the vastness of space, where there is an almost perfect vacuum, celestial objects experience negligible influence from resistive forces like friction or air resistance.

• As a result, objects in motion (e.g., asteroids, comets, space probes) could theoretically continue indefinitely at a constant speed and direction, adhering closely to Newton's First Law, unless acted upon by significant gravitational forces from other celestial bodies.

Basic Physics Concepts

Mass vs. Weight

• Mass: A fundamental intrinsic property of an object that quantifies the amount of matter it contains and its inertia. It does not change regardless of gravitational field or location; always measured in kilograms (kg).

• Weight: The force exerted on an object due to gravity. It varies with the strength of the gravitational field; thus, an object's weight is greater on Earth than on the Moon due to the stronger gravitational pull of Earth.

• Formula for Weight: Weight (W) = mass (m) \times acceleration due to gravity (g), or W = mg.

• Example: A person with a mass of 60 kg has a weight of approximately 60\text{ kg} \times 9.8\text{ m/s}^2 = 588\text{ N} on Earth but experiences less weight on the Moon, where g is approximately 1.62\text{ m/s}^2, resulting in a weight of 60\text{ kg} \times 1.62\text{ m/s}^2 \approx 97.2\text{ N}.

Density

• Definition of Density: Density (\rho) is a derived physical quantity defined as the mass (m) of an object per unit of its volume (V), expressed as \rho = m/V .

• Example Comparison:

  • 1 kg of bricks, made of dense material, has a density of approximately 2000 kg/m³. This means it occupies a relatively small volume.

  • 1 kg of feathers, composed of less dense material with much empty space, has a density of about 2 kg/m³.

• This significant difference in density illustrates that even though their masses are identical (1 kg), 1 kg of feathers takes up significantly more space (volume) than 1 kg of bricks due to its lower density.

Momentum

• Defined as a vector quantity: momentum (p) = mass (m) \times velocity (v), or p = mv (units: kg⋅m/s).

• Momentum depends intrinsically on both the mass of an object and how fast (and in what direction) it is moving.

• Example: A 1 kg brick moving at 1 m/s has a momentum of 1\text{ kg} \times 1\text{ m/s} = 1\text{ kg}{\cdot}\text{m/s}.

  • Increasing either the mass of the object or its velocity (or both) directly increases its momentum. This is why a heavy, fast-moving object is difficult to stop.

Rate of Change

Velocity

• Definition: Velocity (v) is the rate of change of an object's position with respect to time, indicating both its speed and direction. It is measured in meters per second (m/s).

• Example: An object moving at 1 m/s travels 1 meter for every 1 second that passes in a consistent direction.

Acceleration

• Definition: Acceleration (a) is the rate of change of an object's velocity with respect to time. It is a vector quantity, meaning it has both magnitude and direction, and is measured in meters per second squared (m/s²). An object accelerates if its speed changes, its direction changes, or both.

• Example: If an object accelerates at 1 m/s² (meaning its velocity changes by 1 m/s every second), after 1 second it reaches a velocity of 1 m/s (from rest), after 2 seconds it attains 2 m/s, and so forth, assuming constant acceleration.

Newton’s Laws of Motion

Newton's Second Law of Motion

• Definition: This law states that the net force acting on an object is directly proportional to its acceleration and in the same direction as the acceleration. It also states that this net force is equal to the rate of change of momentum of that object over time.

• Formula: The most common expression for Newton's Second Law is:

  F = ma

• Where:

  • F represents the net external force applied to the object (measured in Newtons, N).

  • m is the mass of the object (measured in kilograms, kg).

  • a is the resulting acceleration of the object (measured in meters per second squared, m/s²).

• Example: A mass of 1 kg pushed with a net force of 8 N will experience an acceleration of a = F/m = 8\text{ N} / 1\text{ kg} = 8\text{ m/s}^2.

Newton's Third Law of Motion

• Definition: For every action, there is an equal and opposite reaction. This means that if object A exerts a force on object B (the 'action' force), then object B simultaneously exerts a force of equal magnitude and opposite direction on object A (the 'reaction' force). These forces always act on different objects.

• Example: When walking, your feet push down and back on the ground (action), and the ground simultaneously pushes up and forward on your feet (reaction), propelling you forward.

• Rockets operate based on this principle: they expel high-velocity gas downwards (action), and in response, the expelled gas exerts an equal and opposite force upwards on the rocket, resulting in an upward thrust.

Conservation of Momentum

• Stated that the total momentum of an isolated system (one where no net external forces act) remains constant over time. Momentum can be transferred between objects within the system, but the total sum of momentum before and after an interaction (like a collision) remains unchanged.

• Example: When one billiard ball strikes another, the forces exchanged during the collision cause momentum changes in each individual ball; however, the vector sum of their momenta before the collision is equal to the vector sum of their momenta after the collision, demonstrating that the total momentum is conserved within the system of the two balls.

Angular Momentum

• Defined as the rotational equivalent of linear momentum. For a single particle, it is generally expressed as the product of its mass, its velocity, and its perpendicular distance from the axis of rotation: angular momentum (L) = mass (m) \times velocity (v) \times radius (r), or L = mvr.

• Demonstrated through a figure skater who spins faster when pulling in her arms. By reducing her radius (r) while her mass (m) remains constant, her velocity (v) must increase to conserve her total angular momentum (L), assuming negligible external torques.

• Relationship to Kepler's second law: The conservation of angular momentum is the underlying physical principle behind Kepler's second law. As a planet orbits the Sun, its velocity increases when it is closer to the Sun (smaller r) and decreases when it is farther away (larger r), ensuring that the product mvr (and thus angular momentum) remains constant. This means the planet sweeps out equal areas in equal times.

Newton’s Universal Law of Gravitation

Explanation of Gravitational Force

• Definition: Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is often referred to as an inverse-square law.

• Formula: The mathematical representation of Newton's Law of Universal Gravitation is:

  F = G \frac{m1 m2}{r^2}

• Where:

  • F = the magnitude of the gravitational force between the two objects.

  • G = the universal gravitational constant, approximately 6.674 \times 10^{-11}\text{ N}{\cdot}\text{m}^2/\text{kg}^2, which acts as a proportionality factor.

  • m1 and m2 = the masses of the two interacting objects.

  • r = the distance between the centers of the two objects.

Kepler's Laws of Planetary Motion

• First Law (Law of Ellipses): A planet's orbit is an ellipse with the Sun at one of the two foci. This corrected the ancient belief in perfectly circular orbits.

• Second Law (Law of Equal Areas): A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that a planet moves faster when it is nearer to the Sun (at perihelion) and slower when it is farther away (at aphelion).

• Third Law (Law of Harmonies): The square of a planet's orbital period (T) is directly proportional to the cube of the semi-major axis (a) of its orbit. Mathematically, T^2 \propto a^3. This law allows for the calculation of orbital periods or distances if one is known.

• Newton derived these three empirical laws using his gravitational theory and calculus, demonstrating that they were not merely descriptive rules but direct consequences of the universal law of gravitation and the laws of motion. This unification was a profound scientific achievement.

Summary of Free Fall

• Astronauts experience apparent weightlessness not because there is no gravity in space, but because they are continuously in a state of free fall. They, along with their spacecraft, are constantly falling towards Earth (or another celestial body) at the same rate, essentially orbiting it.

• Weight is the sensation felt when an upward normal force (or tension from a scale) counters the force of gravity, preventing free fall. Since objects in orbit are constantly falling and there's no surface to push back against them (from the perspective of being "in" the spacecraft), the sensation of weight disappears, leading to apparent weightlessness.

Conclusion

• The lecture covered fundamental aspects of Newtonian physics, including Newton's three laws of motion, his universal law of gravitation, the concepts of momentum (linear and angular), and their profound implications in understanding motion both on Earth and throughout the cosmos.

• Reading for review: OpenStax Astronomy, sections 3.2-3.6, with exercises available for further study to reinforce understanding and apply the learned principles.