Preliminary Physics I - Gravitation Lecture 12 - AS Level PDF

PHYS 0100 Preliminary Physics I Ms. Amanda Sankaran Gravitation  After this lecture, students should be able to: Understand the concept of a gravitational field Define gravitational field strength State and use Newton’s Law of universal grav...

PHYS 0100 Preliminary Physics I Ms. Amanda Sankaran Gravitation  After this lecture, students should be able to: Understand the concept of a gravitational field Define gravitational field strength State and use Newton’s Law of universal gravitation Solve problems involving gravitational field Learning strengths at the Earth’s surface or above State the conditions necessary for geostationary orbit Outcomes about the Earth Discuss the motion of geostationary satellites and their applications Solve problems near Earth with geostationary satellites Solve problems involving circular orbits. Gravitational field A gravitational field exists around bodies that have mass. A gravitational field is defined as a region in which a gravitational force acts. It is a region around a body where an object of some significant mass experiences a force when placed in the field. The direction of the gravitational field is the direction a test mass would move if placed anywhere in the field. Gravitational field lines around the Earth If we place an object inside the gravitational field of the Earth, it experiences a force directed towards the center of the Earth. The gravitational field is represented using field lines.  Gravitational field lines around the Earth ▪ The spacing of the field lines gives an idea of the strength of the field. ▪ The closer the field lines, the stronger the field. ▪ The more spaced out the field lines, the weaker the field. ▪ Equally spaced lines represent a uniform field. The Earth’s gravitational field is: 1. Radial 2. Equally spread around the Earth. Gravitational Field on the surface of the Earth On the surface of the earth the gravitational field is approximately uniform. As such we assume the acceleration to due gravity is a constant. 𝒂𝒈 = −𝟗. 𝟖𝟏 𝒎/𝒔𝟐 Gravitational field strength Gravitational field strength is the force per unit mass at that point in the field. 𝑭 𝒈= 𝒎 g – gravitational field strength (N/kg) F – force (N) m – mass (kg) Gravitational field strength is a vector quantity.  Gravitational Field on the surface of the Earth The gravitational force on a mass is called its weight, W. The equation for the gravitational field strength, g: 𝑾 𝒈= 𝒎 Gravitational force Force is required to change the speed or direction of the motion of an object. We observe that an object dropped from a height falls towards the earth. All planets revolve around the Sun. The moon revolves around the Earth. In all these cases there must be some force acting on the objects, planets and the moon. Isaac Newton explained this as gravitational force. Newton’s Theory  Activity to understand the moon’s motion Take a piece of thread and tie a small stone to one end. Hold onto the other end and whirl it around as shown below. Note the motion of the stone. Release the thread and again note the direction of motion of the stone. Before the thread is released, the stone moves in a circular path with certain speed and changes direction at every point. The change in direction involves the change in velocity of acceleration. The force that causes this acceleration and keeps the body moving in a circular path is acting towards the center. This force is called the centripetal force. In the absence of this force, the stones flies off along in a straight line. The straight line will be a tangent to the circular path.  The motion of the moon around the earth is due to the centripetal force. The centripetal force is provided by the force of attraction of the earth. If there were no such force, the moon would pursue a uniform straight-line motion. Why does the earth not move towards the apple? According to Newton’s third law of motion, the apple does not attract the earth. But according to the second law of motion for a given force, acceleration is inversely proportional to the mass of an object. The mass of an apple is negligibly small compared to that of the earth. So we do not see the earth moving towards the apple. Newton concluded that not only does the earth attract an apple and the moon, but all objects in the universe attract each other. This force of attraction between objects is called the gravitational force. Newton’s Law of Gravitation Newton’s Law of Gravitation states that the force of attraction between any two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distances between them.  Newton’s Law of Gravitation Consider two bodies having masses of m1 and m2 respectively and separated by a distance r. 𝑮𝒎𝟏 𝒎𝟐 𝑭=− 𝒓𝟐 The magnitude of the force is: G – Gravitational constant G, the constant of proportionality is called (6.67 x 10-11 𝑚3 𝑘𝑔−1 𝑠 −2 ) the gravitation constant and is 𝑚1 – mass of one body experimentally determined. 𝑚2 – mass of the other body r – distance between the centers of mass of the The minus sign in the equation two bodies indicates that the force is attractive. Gravitational Field Strength The force exerted on m is: 𝑮𝑴𝒎 𝑭=− 𝑹𝟐 Gravitational Field Strength for an object at the Earth’s surface On the Earth's surface the gravitational field strength is g = 9.81 N kg-1 So for an object at the Earth’s surface where the field strength is F = mg and knowing Newton’s 𝑮𝑴𝒎 law of gravitation 𝑭 = − 𝟐 , where we can equate these two: 𝒓 𝑮𝑴𝑬 𝒎 𝒎𝒈 = − 𝑹𝟐𝑬 where 𝑅𝐸 is the radius of the Earth and 𝑀𝐸 is the mass of the earth. So, 𝑮𝑴𝑬 𝒈=− 𝟐 𝑹𝑬  𝑮𝑴 𝒈=− 𝟐 𝑹 Radial Fields 1 𝑔∝ 2 𝑅 Planetary Motion When considering planetary motion, the gravitational force is equal to the centripetal force because the force acts perpendicular to the direction of motion. The earth rotates around the sun. The centripetal force in this circular motion is provided by the gravitational force.  𝑭𝒄𝒆𝒏𝒕𝒓𝒊𝒑𝒆𝒕𝒂𝒍 = 𝑭𝒈𝒓𝒂𝒗𝒊𝒕𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝑚𝑣 2 𝐺𝑀𝑚 = 2 𝑟 𝑟 𝐺𝑀 𝑣2 = 𝑟 2 2𝜋𝑟 𝐺𝑀 = 𝑇 𝑟 4𝜋 2 𝑟 3 Orbital Period 𝑇= 𝐺𝑀  Gravitational potential energy (U) PE = mgh is valid only near the earth’s surface. For objects high above the earth’s surface, an alternate expression is needed 𝑮𝑴𝒎 𝑼=− 𝒓 Zero reference level is infinitely far from the earth, so potential energy is everywhere negative! Energy conservation: 1 𝐺𝑀𝑚 E = KE + U = 𝑚𝑣 2 − 2 𝑟 We define the gravitational potential 𝝓 at a point as the work done in moving unit mass from infinity to that point: 𝑼 𝑮𝑴 𝝓= =− 𝒎 𝒓 Equipotentials An equipotential line is a line drawn through the same gravitational potential. Equipotentials are perpendicular to the field lines Geostationary Orbit A geostationary satellite is one that always appears in the same place in the sky, no matter what the time of day. The conditions for this to occur are: 1. the satellite must have an orbital period of exactly 24 hours 2. the satellite must have a circular orbit above the equator 3. the satellite must be orbiting in the same direction as the Earth is rotating Properties of geostationary orbits Period of orbit around the Earth, T = 24 hours = 86400 s 2𝜋 Angular velocity of orbit, 𝜔 = 𝑇 2𝜋 2𝜋 𝜔= = = 7.27 × 10−5 𝑟𝑎𝑑/𝑠 𝑇 86400 𝑠 Properties of geostationary orbits Radius of orbit of a geostationary satellite, 𝑟 𝐹𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 = 𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝐺𝑀𝑚 𝑚𝜔2 𝑟 = 𝑟2 𝐺𝑀𝑚 𝑚𝜔2 𝑟 = 2 𝑟 𝐺𝑀 𝑟3 = 2 𝜔 3 𝐺𝑀 3 6.67×10−11 ×5.972×1024 𝑟= = = 4.22 × 107 𝑚 𝜔2 7.27 ×10−5 2 Uses of geostationary satellites 1. Weather monitoring 2. Television transmission 3. Telephone communication  Universal Law of Gravitation Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is along the line joining the centers of two objects. Let two object A and B of masses M and m lie at a distance d from each other. Let the force of attraction between the 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. And the force between two objects is inversely proportional to the square of the distance between them.  Value of “G” The SI unit of G can be obtained by substituting the units of force, distance and mass in the above as N m2 kg-2. The value of G was found by Henry Cavendish by using a sensitive balance. The accepted value of G is 6.673 x 10-11 N m2 kg-2. Importance of the Universal Law of Gravitation This law successfully explains several phenomena: ✓ The force that binds us to the Earth. ✓ The motion of the moon around the Earth. ✓ The motion of planets around the Sun. ✓ The tides due to the moon and the Sun. Question a) Define what is meant by a gravitational field. b) Newton’s law of gravitation is given by the equation 𝐺𝑀1𝑀2 𝐹= − Explain what each symbol in the equation 𝑟2. represents. Question a) Define what is meant by a gravitational field. ❑ A region of space ❑ Where a mass experiences a force ❑ Due to the gravitational attraction of another mass 𝐺𝑀1𝑀2 b) Newton’s law of gravitation is given by the equation 𝐹 = Explain what each symbol in the 𝑟2. equation represents. F – gravitational force G – Newton’s gravitational constant (6.67 x 10-11 N m2 kg-2 M1 (or M2) – mass of the point mass(es) r – distance between the center of two (point) masses/ separation of the centers of the two (point) masses. Question c) The Figure below shows the Earth and the moon. Draw arrows on the Figure to show: The gravitational force exerted on the Earth by the moon. Label this force FE. The gravitational force exerted on the moon by the Earth. Label this force Fm. The Figure below shows the Earth and the moon. c) Draw arrows on the Figure to show: i. The gravitational force exerted on the Earth by the moon. Label this force FE. ii. The gravitational force exerted on the moon by the Earth. Label this force Fm. FE Fm Question d) The earth has a mass of 5.97 x 1024 kg and the moon has a mass of 7.35 x 1022 kg. The separation distance between the center of the Earth and moon is 3.84 x 108 m. Calculate the gravitational force between the Earth and the moon. The earth has a mass of 5.97 x 1024 kg and the moon has a mass of 7.35 x 1022 kg. The separation distance between the center of the Earth and moon is 3.84 x 108 m. d) Calculate the gravitational force between the Earth and the moon. List the known quantities: ❑ Mass of the Earth, mE = 5.97 x 1024 kg ❑ Mass of the moon, mm = 7.35 x 1022 kg ❑ Separation distance between Earth and moon, r = 3.84 x 108 m From data and formula sheet: ❑ Gravitational constant, G = 6.67 x 10-11 N m2 kg-2 𝐺𝑀 𝑀 ❑ Force between two masses, 𝐹 = − 1 2 𝑟2 Calculate the gravitational force between the Earth and the moon: − 6.67 × 10 11 × 5.97 × 1024 × 7.35 × 1022 𝐹=− = − 1.98 × 1020 𝑁 3.84 × 108 2 Free Fall Throw a stone upwards. It reaches a certain height and its starts falling down. Whenever objects falls towards the earth under the gravitational force alone, we say that the objects are in free fall. Any change in velocity involves acceleration. This acceleration is called the acceleration due to the gravitational force of the earth (of acceleration due to gravity) denoted by g. The units of g is the same as that of acceleration, that is m s-2 Let the mass of the ball be m and the acceleration due to gravity be, g. From the second law of motion, F = mg 𝑀×𝑚 𝑚𝑔 = 𝐺 𝑑2 𝑀 𝑔=𝐺 𝑑2 M is the mass of the Earth, and d is the distance between the object and Earth. When the object is on or near the surface of the Earth. The distance d will be equal to R, the radius of the Earth. Thus, for objects on or near the surface of the Earth, 𝑀×𝑚 𝑚𝑔 = 𝐺 𝑅2 𝑀 𝑔=𝐺 𝑅2 Activity Drop a stone and paper from a height simultaneously. Observe whether both of them reach the ground simultaneously. The paper reaches the ground little later than the stone. This happened because of air resistance. The air offers resistance due to friction of motion of the falling objects. The resistance offered by air to the paper is more than the resistance offered to the stone. If this experiment is performed in a glass jar from which all the air has been sucked out. The paper and stone would fall at the same rate.  Acceleration experience by an object is independent of its mass. This means that all objects hollow or solid, big or small, should fall at the same rate. g is constant near the earth, all the equations for uniformly accelerated motion of objects become valid with acceleration a replaced by g. Recall ❑Mass of an object is the measure of it’s inertia. ❑The mass of an object is constant and does not change from place to place. ❑Weight of an object is the force with which it is attracted towards the earth. ❑The weight is a force acting vertically downwards; it has both magnitude and direction. Activity Take an empty soft bottle and close it airtight with a stopper. Place it in a bucket filled with water. Push the bottle into the water and you feel an upward push. This indicates that the water exerts an upward force on the bottle in the upward direction. When the bottle is immersed, the upward force exerted by the water on the bottle is greater than its weight, therefore it rises up when released. Buoyancy The upward force exerted by the bottle is known as upthrust or buoyant force. All objects experience a force of buoyancy when they are immersed in a fluid. The magnitude of this buoyant force depends on the density of the fluid. Activity Take a beaker filled with water and place a cork and an iron nail of equal mass on the surface of the water. Observe what happens. The cork floats, nail sinks. This happens because of their difference in densities. The density of the cork is less than the density of water (upthrust of water on cork > weight of the cork) – floats. The density of the iron nail is more than the density of the water (upthrust of water on the iron nail < the weight of the nail) – sinks. Objects of density less than that of the liquid float on the liquid and objects of density greater than that of the liquid will sink in the liquid. Archimedes’ Principle When a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it. Archimedes’ Principle has many applications. Designing of ships and submarines. Lactometers, which are used to determine the purity of milk. Hydrometers used for determining the density of liquids. Relative Density The density of a substance is defined as mass of a unit volume. Density is measured in kilogram per meter cube (kgm-3) The density of a given substance, under specified conditions, remains the same. It is convenient to express density of a substance in comparison with that of water. The relative density of a substance is the ratio of its density to that of water: 𝑫𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝒂 𝒔𝒖𝒃𝒔𝒕𝒂𝒏𝒄𝒆 𝑹𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 = 𝑫𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝒘𝒂𝒕𝒆𝒓 Since the relative density is a ratio of similar quantities, it has no units.