Gravity and Coulomb s Law

Transcription

1 AP PHYSICS 1 Gravity and Coulomb s Law 016 EDITION Click on the following link or scan the QR code to complete the evaluation for the Study Session Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 1

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3 Pre-Assessment Questions P1. Suppose that two pieces of clay with equal mass M are formed into small spheres and set in space a distance R apart. The gravitational force with which these two objects attract is F G. Suppose half of the mass of one sphere is taken and attached to the other sphere so that both pieces of clay are still sphereshaped and still separated by a distance R. Does this increase or decrease the strength F G? Increase Decrease Remain the same Explain your answer. P. The planet Saturn has a greater mass than Earth, and also a greater radius than Earth. Does Saturn have a stronger gravitational field at its surface than Earth, or a weaker gravitational field? Stronger g-field Weaker g-field Not enough information to tell Explain your answer. P3. A neutron star has much more mass than Earth and a radius that is much less than Earth. Does a neutron star have a stronger or weaker gravitational field at its surface as compared to Earth? Stronger g-field Weaker g-field Not enough information to tell Explain your answer. P4. Consider two spherical balloons of equal mass and radius that are in contact with each other. The gravitational force that each balloon exerts on the other is F G. Suppose the two balloons are inflated so that each has double the radius. Ignoring the mass of the air inside the balloons, how will the strength of F G change? P5. Now suppose that the balloons have equal mass and are in outer space. Each balloon has a million extra electrons. The balloons attract each other with a force F G and repel each other with a force F E. If F G balances F E for both balloons, what mass does each balloon have (or is there not enough information to answer this question)? Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 1

4 Universal Gravitation Every object in the universe pulls on every other object in the universe with a gravitational force that is proportional to the product of the two objects masses and inversely proportional to the square of the distance between their centers. This is Newton s Law of Universal Gravitation. Gm1m F G = r G = Universal Gravitational Constant = Nm /kg m 1, m = Any two masses that attract each other r = Separation distance between the objects centers We only use F g = mg for the force of gravity when the object (m) is near the surface of a planet (having gravitational field g). But Universal Gravitation calculates the force of gravity for any two objects, such as a planet and star (the planet is not near the surface of the star, hopefully). Just as there is a universal equation for gravitational force, there is a universal equation for gravitational potential energy. We only use U g = mgy for the gravitational potential energy when the object (m) is near the surface of a planet (having gravitational field g) and the object has a height (y, measured from any origin you choose). However, if we are discussing a planet and star or planet and moon or two stars or two planets (where one object is not near the surface of the other object), we must use this equation: Gm1m = r U G G = Universal Gravitational Constant = Nm /kg m 1, m = Any two masses that attract each other r = Separation distance between the objects centers Note: The gravitational potential energy from this equation is always negative, since all four variables are always positive. This equation almost looks like the force equation, but the r is not squared. Both equations appear on the AP Physics Table of Equations; do not mix them up! If the two objects come closer together, then the potential energy goes down (r decreases, making the fraction bigger, but a bigger negative is a lower number). If the two objects move farther apart, then the potential energy goes up (r increases, making the fraction smaller, but a smaller negative is a higher number). Orbits If a large object (the central body having mass M) pulls a smaller object (the satellite having mass m) into a periodic circular or elliptical motion, then the satellite orbits the central body. All orbits obey the laws of conservation of energy and angular momentum, but these laws are only worth considering for elliptical orbits. Circular orbits have an equation where the gravitational force is treated as the centripetal force. Primary Equation(s) Important to Know Circular Orbits Set gravitational force as the centripetal force: GmM mv = r r Set speed equal to circumference over period: v = πr / T As M increases, v increases and/or T decreases. As r increases, v decreases and/or T increases. The mass m affects nothing about the orbit. Elliptical Orbits Initial mechanical energy = final mech. energy: 1 GmM 1 GmM mvi = mv f ri rf Initial angular momentum = final angular mom.: mv r sin θ = mv r sinθ i i i As m gets closer to M, speed and kinetic energy increase, and potential energy decreases (into a deeper negative). Total mechanical energy of an orbit must be negative (or there is no orbit). f f f Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at

5 Coulomb s Law Just as objects with mass pull on each other with a force called gravity, every object in the universe that has a net amount of charge pulls or pushes on every other object in the universe that has a net charge. Net charge means has more of one sign of charge than the other. The signs of charge are positive and negative. Opposite-sign net charges attract each other, but like-sign net charges repel each other. This is the electrostatic force. The strength with which two charged objects attract or repel is given by Coulomb s Law. This law looks very much like Universal Gravitation because it involves a universal constant, the multiplication of two objects properties (charge here instead of mass), and the force is inversely proportional to the square of the separation distance. F E = kq q 1 r k = Coulomb s Law Constant = Nm /C q 1, q = Any two net charges (charge is measured in Coulombs, C) r = Separation distance between the objects centers A coulomb of charge is a humongous amount of charge. When you rub your hair on a balloon and cause it to feel tingly, the balloon actually takes electrons off of your hair. Electrons are negatively charged, so if the balloon has extra electrons, then the balloon has a negative net charge. But the amount of charge you build up is in the billionths of a coulomb (10 9 C). Whole coulombs of charge are only collected in large-scale structures like thunderclouds (right before a lightning strike). Conservation of Charge When we think of conservation, we think cannot be created or destroyed. This is misleading, as there are several ways to create charge or make it go away. The key to conservation is the net amount before equals the net amount after. Conservation of charge states that any charging process produces or defeats equal amounts of both signs of charge, so that the net charge of a closed system is the same before and after the process. Go back to the balloon and your hair. Suppose that the system is your living room, with you, the balloon, a TV, and a flashlight that is turned on in it. The living room starts with about 10 nc of charge (nc = nanocoulombs = billionths of a coulomb) because the Earth s surface has about 1 nc on it for every square meter. You charge the balloon so that the balloon has 0 nc. The net charge of the living room ( 10 nc) didn t change. The flashlight doesn t gain or lose charge even though it is on charges are just running around the flashlight s circuit like a race. The TV, even though it is plugged in, has charges entering one prong at the same time that charges leave the other prong. The flashlight, TV, and room don t change charge, but if the balloon gains 0 nc of charge, your hair must now gain +0 nc of charge so that the entire system s net charge is the same before and after the charging process. The Relative Strength of the Gravitational and Electrostatic Forces The electrostatic force is far, far, far, far stronger than the gravitational force. Period. Try this for yourself: two protons (mass = kg, charge = C) are separated by m in the nucleus of a helium atom. Plugging into Universal Gravitation and Coulomb s Law, we get F G = N, and F E = 30 N. Let me write those numbers out for you: F E = 30 N F G = N If F G were represented by a 1 cm long arrow, then F E would be an arrow that points a million times longer than the distance to the edge of the observable universe. Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 3

6 Multiple-Choice Questions M1. Sphere 1 has mass M 1 and positive charge q 1. Sphere has a much smaller mass m and a positive charge q. It is not known which sphere has greater charge, but it is known that the masses and charges are related by the statement GM 1 m = kq 1 q. The spheres are isolated in space, far from other objects. Sphere 1 is at rest and sphere initially moves to the right so as to pass above sphere 1. Which diagram below shows the path that sphere will take due to the gravitational and electrostatic interactions between the two spheres? (A) (B) Questions 3 4: Two metal spheres of identical size exist in an isolated region of space such that they only experiences forces from each other. One sphere has mass M 1 and charge positive charge Q 1 and the other has mass M and positive charge Q. M3. Suppose that the masses are unknown, but it is known that Q 1 is much greater than Q and that both spheres have balanced electrostatic and gravitational forces acting on them. If a thin metal wire connects the two masses for a short instant, what will the masses do? (A) Move together due to the increased gravitational force (B) Move together due to the decreased electrostatic force (C) Move apart due to the decreased gravitational force (D) Move apart due to the increased electrostatic force (C) (D) There is no way to make a determination without knowing which sphere has greater charge. M4. Suppose instead that both objects have identical positive charges, but that M 1 is much greater than M. When both spheres are released from rest, sphere moves away while sphere 1 appears to remain at rest. How will the speed and magnitude of acceleration of sphere change as sphere moves away from sphere 1? M. Two planets A and B orbit the same star in circular orbits. Planet A has an orbital radius R and speed v. Which of the following could be the orbital radius and speed of planet B? (A) R and v/4 (B) 4R and v/ (C) 4R and v/8 (D) 8R and v/4 Speed (A) Increase (B) Increase (C) Decrease (D) Decrease Acceleration Increase Decrease Increase Decrease Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 4

7 M5. Spheres 1 and both carry positive charge and are connected by insulating strings to a ceiling. When the two spheres hang in equilibrium, they hang at the same height, but sphere 1 s string is more vertical, as shown in the diagram above. Based on this observation, it can be concluded that, compared to sphere, sphere 1 has (A) more charge (B) less charge (C) more mass (D) less mass M6. An astronomer makes the following measurements of the planet Jupiter and one of its moons, Ganymede. Which two measurements are required to determine Jupiter s mass? Select two answers. (A) The time Jupiter takes to orbit the Sun once. (B) The average distance between Jupiter and the Sun. (C) The time Ganymede takes to orbit Jupiter once. (D) The average distance between Ganymede and the Jupiter. Free-Response Questions F1. Somewhere in space far from other masses or charges, two extremely dense spherical objects 1 and are initially at rest and have centers separated by 3.0 meters. The masses of the two spheres are m 1 = kg and m = kg and the charges are q 1 = C and q = C. (a) Calculate F G, the gravitational force acting on sphere 1, and F E, the electric force acting on sphere 1. (b) The diagram below shows the two spheres set on a gridded background. On each sphere, draw and label vectors representing the gravitational force F G and electric force F E acting on each sphere. Use the grid to draw all four vectors with lengths that reflect the relative strengths of each force. (c) A student analyzing the situation notices that, when the spheres are 3.0 m apart, the electric force is twice as strong as the gravitational force. How far apart must the spheres be separated in order for the gravitational and electric forces to balance? Justify your answer. Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 5

8 F. Two identical metal rods are given equal positive charges Q and coated in a thin insulating material that would prevent any charge from being transferred to or away from either rod. A student wishes to determine the relationship between the electric force acting on each rod and the separation distance between the rod s axes. The student resolves to only perform experiments in which the two rods are set parallel to each other. (a) Outline a procedure the student could follow in order to make measurements that would allow the student to determine the relationship between the electric force acting on the rods and the separation distance between their axes. Be sure to mention any equipment the student is to use and how that equipment is to be used. (b) Explain the process of data analysis. How can the measurements be used to determine the type of relationship between electric force and separation distance? Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 6

9 (c) The student consults a physics textbook and finds that the electric force between the two rods in this kq situation is given by the equation F E =, where k is the coulomb s law constant, r is the separation rl between axes, and L is the length of each rod. The student s data is shown in the table below: Separation r (m) Electric Force F E (N) Is the data taken consistent with the equation? Explain your reasoning. (d) Suppose that the rods are cylindrical in shape, and that the student holds the two rods on a flat table so that they are very close to each other and parallel to each other. The student releases the rods, and the rods exert a repulsive electric force on each other so that both rods roll away in opposite directions. Assume that the only unbalanced force acting on the two rods is the electric force of repulsion. i. Does the magnitude of the acceleration of each rod increase, decrease, or remain the same as the rods roll away from each other? Justify your answer. ii. Does the speed of each rod increase, decrease, or remain the same as the rods roll away from each other? Justify your answer. Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 7

10 F3. A student observes a star of mass M with a gas cloud of total mass m orbiting it in a circular orbit of radius R. The gas cloud is able to maintain a nearly-spherical shape under its own gravity, and can be treated as a single object. The direction of the orbit is counterclockwise. (a) Write expressions for the following in terms of G, M, m, and R only. i. The speed that the gas cloud orbits the star ii. The total energy of the star-cloud system, assuming that neither the star nor cloud rotate iii. The angular momentum of the orbiting cloud taken about the center of the star Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 8

11 (b) The student makes several observations over a long period of time. During the interval over which the student makes observations, forces that are completely internal to the system cause half of the mass of the gas cloud to spiral into the star (a process called accretion ). The student observes that, after the accretion event, the gas cloud still orbits the star in a circle of radius R. For each of the following parts, answer the question and use your answers to part (a) to justify your answers. i. Is the orbital speed of the remaining gas faster, slower, or the same speed as before? ii. Does the mechanical energy of the star-cloud system increase, decrease, or remain the same as a result of the accretion event? iii. The star was not rotating before the accretion event; is the star now rotating clockwise, counterclockwise, or still not rotating? Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 9

12 F4. Spheres 1 and each carry amounts of charge and have masses that are not necessarily equal. They are connected to the same point in a ceiling by strings of different length and remain at rest in equilibrium as shown in the left diagram labeled Case A. Later, sphere is removed and sphere 3 (also having some charge) is set in its place, and the lengths of the strings are adjusted so that the system remains at rest in equilibrium as shown in the right diagram labeled Case B. Both diagrams are to scale and have a grid in the background to assist in estimating the relative angles and lengths of the strings. Case A Case B (a) The four circles below represent the spheres (1 and in case A, and 1 and 3 in case B). For all of the circles, draw and label the weight force W, tension force T, and electrostatic force E acting on the sphere. Use the grid in the background to draw the forces with correct angles and relative components. Do not draw the components on any force vector. Case A Case B (b) The three spheres have different masses and different charges. i. In the spaces shown, rank the spheres in terms of their mass: > > ii. Is it possible to rank the spheres in terms of their charges? Yes No iii. In a well-organized, paragraph-length response, justify your answers to (b)-i and (b)-ii. Your response may reference the above diagrams or include additional diagrams or equations. Copyright 016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at 10

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