PHY101 General Physics I 2024-2025 PDF

PHY101- GENERAL PHYSICS I (2 UNITS) Module 1 Gravitation: Newton’s Law of Gravitation Module 2 Kepler’s, Laws of Planetary Motion Module 3 Gravitational Potential Energy and Escape velocity Module 4 Satellites motion and orbits Module 1 Gravitation: Newton’s Law of...

PHY101- GENERAL PHYSICS I (2 UNITS) Module 1 Gravitation: Newton’s Law of Gravitation Module 2 Kepler’s, Laws of Planetary Motion Module 3 Gravitational Potential Energy and Escape velocity Module 4 Satellites motion and orbits Module 1 Gravitation: Newton’s Law of Gravitation Newton’s Law of Gravitation Newton's law of gravitation states that everybody in this universe attracts every other body with a force, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The direction of the force is along the line joining the particles. Mathematical derivation: The direction of force is along the line joining the centers of two objects. Let two objects A and B of masses M and m lie at a distance d from each other as shown in figure. Let the force of attraction between two objects be F. According to the universal law of gravitation, the force between two objects is directly proportional to the product of their masses. That is, And the force between two objects is inversely proportional to the square of the distance between them, that is, Combining equation (i) and (ii) 1 Where G is the constant of proportionality and is called universal gravitation constant. The SI unit of G can be obtained by substituting the unit of force, distance and mass in equation (iii). Some importance facts of Universal Law of Gravitation The universal law of gravitation successfully explained several phenomena which were believed to be unconnected: the force that binds us to the earth; the motion of the moon around the earth; the motion of planets around the Sun; and the tides due to the moon and the Sun. Gravitational force between light objects and heavy objects: The formula applied for calculating gravitational force between light objects and heavy objects is the same, i.e. 2 Let us take three cases: 1. When two bodies of mass 1 kg each are 1 metre apart. This is extremely small. Hence, we conclude that though every pair of two objects exerts gravitational pull on each other, yet they cannot move towards each other because this gravitational pull is too weak. 2. When a body of mass 1 kg is held on the surface of Earth. Gravitational force of attraction between the body and Earth, It means that the Earth exerts a gravitational force of 9.8 N on a body of mass 1 kg. This force is much larger as compared to the force when both the bodies are lighter. That is why when a body is dropped from a height, it falls to the Earth. 3. When both the bodies are heavy. Let us calculate gravitational force of attraction between Earth and the Moon. 3 The gravitational force between Earth and Moon, This is really large. It is this large gravitational force exerted by Earth on Moon, which makes the Moon revolve around the Earth. Question 1 Two persons having mass 100kg each are standing such that the centre of gravity is 1m apart. Calculate the force of gravitation and also calculate the force of gravity on each. Force of gravity, Force of gravity, F’ is much greater than F so the persons will not move towards each other but each of them moves towards the earth. GRAVITATIONAL FORCE BETWEEN DIFFERENT OBJECTS: Gravitational Force between Sun and Earth: 4 Gravitation force between the sun and the earth, The gravitational force between the sun and the earth is very large (i.e. 3.6 × 1022 N). This force keeps the earth bound to the sun. Gravitational Force between Moon and Earth: Gravitational force between the earth and the moon, This large gravitational force keeps the moon to move around the earth. This large gravitational force is also responsible for the ocean tides Question 2 Two bodies A and B having mass 4m and 8m respectively are kept at a distance d apart. Where a small particle should be placed so that the net gravitational force on it due to the bodies A and B is zero? 5 Solution: it is clear that the particle must be placed on the line AB, suppose it is at a distance x from A. Let its mass is m’. The force on m’ due to A, Figure 1 6 Gravitational Potential Energy Gravitational potential energy is the energy an object has due to its position above Earth, energy due to its height. We know this energy exists because it takes effort to lift an object up to a height and also because when we release an object, it falls, gaining kinetic energy. The gravitational potential 𝐺𝐺𝐺𝐺 𝑈𝑈 = 𝑟𝑟 Or the gravitational potential energy is given as 𝐺𝐺𝐺𝐺𝐺𝐺 𝑈𝑈 = 𝑟𝑟 M is the source mass placed along the x-axis. m is the test mass at infinity. r is the distance from the source mass at which the gravitational potential energy is determined. Escape Speed If you fire a projectile upward, usually it will slow, stop momentarily, and return to Earth. There is, however, a certain minimum initial speed that will cause it to move upward forever. This minimum initial speed is called the (Earth) escape speed. Consider a projectile of mass m, leaving the surface of a planet (or some other astronomical body or system) with escape speed v. The projectile has a kinetic energy K given by and a potential energy U given by the eqn. 1 𝐺𝐺𝐺𝐺 𝐾𝐾. 𝑒𝑒 = 2 𝑚𝑚𝑣𝑣 2 = 𝑟𝑟 where the gravitational potential 2𝐺𝐺𝐺𝐺 𝑉𝑉 = 𝑟𝑟 is called the escape velocity. 𝐺𝐺𝐺𝐺 Note that we have 𝑔𝑔 = 𝑟𝑟 2 where we have 𝑉𝑉 = 2𝑔𝑔𝑔𝑔 7 If the average radius of the Earth is , then 𝑉𝑉 = √2 × 9.8 × 6 × 103 = 11.2 𝐾𝐾𝑠𝑠 −1 Any particle with velocity greater than this will escape from the atmosphere of the Earth What is explained by Kepler's laws of planetary motion? They describe how (1) planets move in elliptical orbits with the Sun as a focus, (2) a planet covers the same area of space in the same amount of time no matter where it is in its orbit, and (3) a planet's orbital period is proportional to the size of its orbit (its semi-major axis). Kepler’s laws of planetary motion can be stated as follows: Kepler’s First Law – The Law of Orbits According to Kepler’s first law, “All the planets revolve around the sun in elliptical orbits having the sun at one of the foci”. The point at which the planet is close to the sun is known as perihelion (about 147 million kilometres from the sun), and the point at which the planet is farther from the sun is known as aphelion (152 million kilometres from the sun). It is characteristic of an ellipse that the sum of the distances of any planet from two foci is constant. 8 Kepler’s Second Law – The Law of Equal Areas Kepler’s second law states, “The radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time”. As the orbit is not circular, the planet’s kinetic energy is not constant in its path. It has more kinetic energy near the perihelion, and less kinetic energy near the aphelion implies more speed at the perihelion and less speed (v min ) at the aphelion. If r is the distance of the planet from the sun at perihelion (r min ) and at aphelion (r max ), then, r min + r max = 2a × (length of the major axis of an ellipse)....... (1) Using the law of conservation of angular momentum, the law can be verified. At any point of time, the angular momentum can be given as L = mr2ω. Kepler’s second law can also be stated as, “The areal velocity of a planet revolving around the sun in elliptical orbit remains constant, which implies the angular momentum of a planet remains constant”. As the angular momentum is constant, all planetary motions are planar motions, which is a direct consequence of central force. Kepler’s Third Law – The Law of Periods According to Kepler’s law of periods, “The Square of the time period of revolution of a planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major axis”. 9 The shorter the orbit of the planet around the sun, the shorter the time taken to complete one revolution. Using the equations of Newton’s law of gravitation and laws of motion, Kepler’s third law takes a more general form. Where M 1 and M 2 are the masses of the two orbiting objects in solar masses. Summary Questions on Kepler’s Law Q1 What does Kepler’s first law state? According to Kepler’s first law, all the planets revolve around the Sun in elliptical orbits, with the Sun as one of the foci. Q2 What does Kepler’s second law state? According to Kepler’s second law, the speed at which the planets move in space continuously changes. The second law helps to explain that when the planets are closer to the Sun, they will travel faster. Q3 What is Kepler’s third law? Kepler’s third law also called the law of periods, states that the square of the orbital period is proportional to the cube of its mean distance, R. Q4 Why are the orbits of the planets not circular? For the orbits to be circular, it requires the planets to travel with a certain velocity, which is extremely unlikely. If there is any change in the velocity of the planet, the orbit will be elliptical. Satellites motion and orbits 10 Planets and Satellites- Kepler’s laws The motion of satellites, both natural and artificial, is governed by Kepler’s laws: 1. The law of orbits. All planets move in elliptical orbits with the Sun at one focus. 2. The law of areas. A line joining any planet to the Sun sweeps out equal areas in equal time intervals. (This statement is equivalent to conservation of angular momentum.) 3. The law of periods. The square of the period T of any planet is proportional to the cube of the semi major axis a of its orbit. For circular orbits with radius r where M is the mass of the attracting body—the Sun in the case of the solar system. For elliptical planetary orbits, the semi major axis a is substituted for r. Circular motion A satellite orbiting the Earth moves in a circular motion at a constant speed and at a fixed height. The motion is caused by the gravitational force that acts between the Earth and the satellite and centripetal force. The force of gravity acts in a direction perpendicular to the direction of motion of the satellite throughout the trajectory. Some satellites follow the rotation of the Earth and move from west to east, while others have orbits taking them over the poles and travel north to south or south to north. The Gravitational Force plays a huge role in satellite technology. Without Motion of satellites in space it is not possible for us to use satellite for communication. Let’s first understand what a satellite is. An object orbiting around any celestial body (planet, moon, sun, asteroid etc.) is known as the satellite. Motion of Satellites Satellites follow a circular route as they orbit the planet which is known as Motion of Satellites. The satellite’s velocity is shown by the tangent to this circular route, whereas acceleration is in the direction of the circle’s centre. Between the satellite and the Earth, a number of factors interact to cause them to orbit in a circular pattern. To learn in depth about Motion of satellite we need to study about critical and escape velocity of a satellite. 11 Kinetic, Potential, Total and Binding Energy of a Satellite Total Energy of a Satellite in Earth’s orbit The kinetic energy in a moving satellite around the earth due to its orbital motion is given as Now as explained earlier vc is the critical velocity of a satellite and the potential energy of a satellite is given as GMm U =− r Hence the total energy of a satellite revolving around the earth is equal Kinetic energy and potential energy of the satellite Now if the satellite is orbiting close to the earth’s surface, the r = R Binding Energy of a Satellite Binding Energy of a Satellite: The minimum energy required to remove the satellite from Earth’s gravitational field of influence. since the total energy of a satellite GMm E=− 2R Here negative sign shows that the satellite is bounded around the earth now if we want to remove any satellite from earth’s field of influence, we must provide energy equals to or greater than GMm E=+ 2R This means that the Minimum binding energy of a satellite, 12  GMm B.E = + 2R We discussed the above different energies and their equations for a satellite’s motion under gravitational force. Now it’s also important to remember the relation between these different types of energies. For the table below, we have Kinetic Energy = K Potential Energy = U Total Energy = E Binding Energy = B.E. Short Questions 1. What is Critical Velocity? Critical velocity is defined as the minimum velocity required to put the satellite in a stable circular orbit around any celestial object. 2. What is Escape Velocity? Escape velocity is defined as the minimum velocity required by the satellite to project it vertically outward in order to escape Earth’s Gravitational field of influence. 3. What is Satellite? An object orbiting around any celestial body (planet, moon, sun, asteroid etc.) is known as a satellite. 4. What is a Natural Satellite? A celestial object orbiting around the sun or planet is known as a Natural satellite. 5. What is a Manmade Satellite? A man celestial object put in an orbit around in space is known as an artificial satellite. 13