Kepler's Laws of Planetary Motion

Kepler's Laws

• Planetary motion follows three laws described by Johannes Kepler, known as Kepler's laws
• Kepler's laws were derived from studying data on planetary positions and motion
• Kepler's First Law: The orbit of a planet is an ellipse with the Sun at one of the foci
• Kepler's Second Law: The line joining the planet and the Sun sweeps equal areas in equal intervals of time
• Kepler's Third Law: The square of the period of revolution of a planet is directly proportional to the cube of the mean distance of the planet from the Sun

Newton's Universal Law of Gravitation

• Newton formulated his theory of Universal Gravity, which states that every object in the Universe attracts every other object with a definite force
• The force is directly proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between them
• The gravitational force can be expressed mathematically as F = G * (m1 * m2) / d^2, where G is the Universal Gravitational constant

Introduction to Scientists

• Johannes Kepler (1571-1630) was a German astronomer and mathematician who discovered Kepler's laws
• Sir Isaac Newton (1642-1727) was an English scientist who formulated Newton's laws of motion and the law of Universal Gravitation

Force and Motion

• A force is necessary to change the speed or direction of motion of an object
• Newton's laws of motion and the law of Universal Gravitation help explain the behavior of objects on Earth and in the Universe

Circular Motion and Centripetal Force

• An object moving in a circle experiences a force directed towards the centre of the circle, known as the centripetal force
• The centripetal force is necessary to keep the object moving in a circle and not flying off in a straight line
• Examples of circular motion include the moon orbiting the Earth and the Earth orbiting the Sun

Gravitation

• Gravitation is a universal force that acts between all objects with mass
• The force of gravitation is responsible for the motion of planets, stars, and galaxies
• The force of gravitation is also responsible for the falling of objects on Earth

Centripetal Force and Uniform Circular Motion

• The centripetal force is related to the speed and radius of the circle by the equation F = m * v^2 / r
• The force of gravitation provides the centripetal force necessary for uniform circular motion
• The speed of an object in uniform circular motion can be expressed in terms of the period of revolution and the radius of the circle

Solved Examples

• Example 1: Calculating the gravitational force between two objects
• Example 2: Calculating the velocity of an object in uniform circular motion### Mahendra's Acceleration
• Mahendra's mass is 75 kg, and the force applied to him is 733 N.
• Mahendra's acceleration is calculated as 9.77 m/s² using the formula a = F/m.
• After 1 second, Mahendra's velocity is 9.77 m/s.

Newton's Law of Gravitation

• Every object attracts every other object with the same force.
• The gravitational force due to the earth acts on the moon and artificial satellites, causing them to orbit the earth.

Earth's Gravitational Acceleration

• The earth exerts a gravitational force on objects near it, resulting in acceleration.
• This acceleration is called acceleration due to gravity, denoted by 'g', and is directed towards the center of the earth.
• The value of g on the surface of the earth is 9.77 m/s².

Value of g on the Surface of the Earth

• The value of g can be calculated using Newton's universal law of gravitation.
• The formula for g is g = GM/r², where G is the gravitational constant, M is the mass of the earth, and r is the distance from the center of the earth.
• The value of g on the surface of the earth is 9.77 m/s².

Variation in Value of g

• The value of g changes along the surface of the earth due to the earth's slightly ellipsoidal shape.
• The value of g is highest at the poles (9.832 m/s²) and lowest at the equator (9.78 m/s²).
• The value of g also changes with height, decreasing as the distance from the center of the earth increases.
• The value of g changes if we go inside the earth, decreasing as the distance from the center of the earth decreases.

Change of g with Height and Depth

• The value of g decreases with increasing height, but the decrease is small for heights compared to the earth's radius.
• The value of g also changes with depth, decreasing as we go deeper inside the earth.

Mass and Weight

• Mass is the amount of matter present in an object, measured in kilograms (kg).
• Mass is a scalar quantity and does not change, even when going to another planet.
• Weight is the force with which the earth attracts an object, measured in Newtons (N).
• Weight is a vector quantity and changes from place to place, depending on the value of g.
• Colloquially, weight is often used to refer to mass, but scientifically, weight refers to the gravitational force on an object.

Universal Gravitation

• Gravity is a universal force, acting between every point mass, attracting them to each other along the line intersecting both points.
• This force is a two-way interaction, where both objects attract each other with the same force, but in opposite directions.
• Compared to other fundamental forces, gravity is much weaker, but it can act over vast distances, even across millions of kilometers.

Gravitational Forces

• The gravitational force (F) between two objects is calculated by the equation: F = G * (m1 * m2) / r^2, where G is the gravitational constant (6.67408e-11 N*m^2/kg^2).
• The masses of the two objects are represented by m1 and m2, and r is the distance between their centers.
• A gravitational field is a region around an object where the gravitational force can be detected, and it is a vector field surrounding every object with mass.
• Gravitational potential energy (U) is the energy an object has due to its position within a gravitational field, calculated by the equation: U = m * g * h.
• The variables in this equation are the mass of the object (m), the acceleration due to gravity (g, approximately 9.8 m/s^2 on Earth), and the height of the object above the ground (h).