Uwu BBST 171-2 Gravitation PDF

GRAVITATION Little-bit of History In the 4th century BC, the Greek philosopher Aristotle believed that there is no effect or motion without a cause. The cause of the downward motion of heavy bodies, such as the element earth, was related to their nature, which caused them to move downward toward th...

GRAVITATION Little-bit of History In the 4th century BC, the Greek philosopher Aristotle believed that there is no effect or motion without a cause. The cause of the downward motion of heavy bodies, such as the element earth, was related to their nature, which caused them to move downward toward the center of the universe, which was their natural place. Little-bit of History Conversely, light bodies such as the element fire, move by their nature upward toward the inner surface of the sphere of the Moon. Thus in Aristotle's system heavy bodies are not attracted to the earth by an external force of gravity, but tend toward the center of the universe because of an inner gravitas or heaviness. Little-bit of History During the 17th century, Galileo found that, counter to Aristotle's teachings, all objects accelerated equally when falling. In the late 17th century, as a result of Robert Hooke's suggestion that there is a gravitational force which depends on the inverse square of the distance Little-bit of History Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler's third law was published in 1619. In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun. Kepler’s Laws 1. The orbit of a planet is an ellipse with the Sun at one of the two foci. 2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. 3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. Kepler’s Laws 1. The orbit of a planet is an ellipse with the Sun at one of the two foci. Kepler’s Laws 2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. Kepler’s Laws 3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. 2 3 𝑇 ∝𝑟 Little-bit of History Isaac Newton was able to mathematically derive Kepler's three kinematic laws of planetary motion in 1666 "I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centers about which they revolve, and thereby compared the force requisite to keep the moon in her orb with the force of gravity at the surface of the earth and found them to answer pretty nearly." — Isaac Newton, 1666 Newton's Law of Gravitation So Newton's original formula was: 𝑀𝑚 𝐹∝ 2 𝑟 Newton's Law of Gravitation states that the gravitational force 𝐹 between two point masses 𝑀 and 𝑚 a distance 𝑟 apart is attractive, acts along the line joining their centers, and is proportional to the masses and inversely proportional to the square of their separations. Newton's Law of Gravitation In the SI system, the constant of proportionality is 𝐺 , the gravitational constant, which has a value of, −11 2 −2 6.67 × 10 N 𝑚 𝑘𝑔 This gravitational constant was first measured in 1797 by Henry Cavendish. So we may write the equation as, 𝑀𝑚 𝐹=𝐺 2 𝑟 Newton's Law of Gravitation Examples 1. Assuming a planet orbiting circularly around the sun, show that the square of the orbital period of a planet is proportional to the cube of the radius of its orbit. 2. Assuming the earth with uniform circular motion around the sun calculate the mass of the sun. Radius of the circular orbit 𝑟𝑠 = 1.5 × 1011 𝑚. Gravitational Field Gravitational Field A vector field is an assignment of a vector to each point in a subset of space. We can assign a gravitational force(vector) to each point in a space. This vector field is known as the gravitational field. The gravitational field defined as the gravitational force exerted per unit mass at a point in the field. Gravitational Field It is a vector field, and points in the direction of the force that a small test mass would feel at that point. For a point particle of mass 𝑀 , the magnitude of the resultant gravitational field strength 𝑔, at distance 𝑟 from 𝑀, is 𝐺𝑀 𝑔= 2 𝑟 Gravitational Field The gravitational force acting on a mass 𝑚, which is also sometimes described as its weight in the gravitational field 𝑔, is given by, 𝐺𝑀𝑚 𝐺𝑀 𝐹 = 2 = 𝑚 2 = 𝑚𝑔 𝑟 𝑟 Example −2 Using 𝑔 = 9.8𝑚𝑠 , 𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑎𝑟𝑡ℎ 𝑅 = 6.4 × 106 𝑚 and 𝐺 = 6.7 × 10−11 𝑁𝑚2 𝑘𝑔−2 Calculate the Mass of the Earth and Density of the Earth. Inertial Mass : The inertial mass of an object determines its acceleration in the presence of an applied force. According to Newton's second law of motion, if a body of fixed mass 𝑚 is subjected to a force 𝐹, its acceleration 𝑎 is given by 𝐹/𝑚. Applied The inertial 𝐹 force on the object mass of an 𝑚= object 𝑎 Its acceleration Gravitational Mass : A body's mass also determines the degree to which it generates or affected by a gravitational field. The Attractive gravitational 𝐹 Force mass of an 𝑚= object 𝑔 gravitational field strength Generally, 𝑚𝑖𝑛𝑒𝑟𝑡𝑖𝑎𝑙 ≠ 𝑚𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 Gravitational Potential Energy The gravitational potential energy (GPE) of a body of mass m in a gravitational field is the energy stored as a result of the body's position in the field. If the field is due to a uniform spherical mass M, and the particle is outside of M at a distance r from the center of M, then the particle's gravitational potential energy is conventionally given by, 𝐺𝑀𝑚 𝑈𝑔𝑟𝑎𝑣 = − 𝑟 Gravitational Potential The gravitational field potential due to a uniform spherical mass M, outside of M at a point a distance r from the centre of M, is given by 𝐺𝑀 𝑉𝑔𝑟𝑎𝑣 = − 𝑟 Example In a region close to the surface of the Earth, the gravitational acceleration, g, can be considered constant. Show that the difference in potential energy from one height to another can be approximate as 𝑚𝑔∆ℎ.