Introduction to Potential Energy Questions

Introduction to Potential Energy Learning Goal: Understand that the work done by conservative forces is independent of the path taken, and that these conservative forces can be incorporated into a new form of energy called potential energy that must be added to the kinetic energy to get the total mechanical energy. The first part of this problem contains short-answer questions that review the work-energy theorem. In the second part we introduce the concept of potential energy. But for now, please answer in terms of the work-energy theorem. Work-Energy Theorem The work-energy theorem states K f = K i + W net, where W net is the total work done by all forces that act on the object, and K iK_i and K f K_f are the initial and final kinetic energies, respectively. Part A The work-energy theorem states that a force acting on a particle as it moves over a ______ changes the ______ energy of the particle if the force has a component parallel to the motion. Choose the best answer to fill in the blanks above: ANSWER: distance / potential distance / kinetic vertical displacement / potential none of the above Correct It is important that the force have a component acting in the direction of motion. For example, if a ball is attached to a string and whirled in uniform circular motion, the string does apply a force to the ball, but since the string's force is always perpendicular to the motion it does no work and cannot change the kinetic energy of the ball. Part B To calculate the change in kinetic energy, you must know the force as a function of _______. The work done by the force causes the kinetic energy change. Choose the best answer to fill in the blank above: ANSWER: acceleration work position potential energy Correct Part C To illustrate the work-energy concept, consider the case of a stone falling from x ix_i to x f x_f under the influence of gravity. Using the work-energy concept, we say that work is done by the gravitational _____, resulting in an increase of the ______ energy of the stone. Choose the best answer to fill in the blanks above: ANSWER: force / kinetic potential energy / potential force / potential potential energy / kinetic Correct Potential Energy You should read about potential energy in your text before answering the following questions. Potential energy is a concept that builds on the work-energy theorem, enlarging the concept of energy in the most physically useful way. The key aspect that allows for potential energy is the existence of conservative forces, forces for which the work done on an object does not depend on the path of the object, only the initial and final positions of the object. The gravitational force is conservative; the frictional force is not. The change in potential energy is the negative of the work done by conservative forces. Hence considering the initial and final potential energies is equivalent to calculating the work done by the conservative forces. When potential energy is used, it replaces the work done by the associated conservative force. Then only the work due to nonconservative forces needs to be calculated. In summary, when using the concept of potential energy, only nonconservative forces contribute to the work, which now changes the total energy: K f + U f = E f = W nc+ E i = W nc+ K i + U i,where U f U_f and U iU_i are the final and initial potential energies, and W ncW_nc is the work due only to nonconservative forces. Now, we will revisit the falling stone example using the concept of potential energy. Part D Rather than ascribing the increased kinetic energy of the stone to the work of gravity, we now (when using potential energy rather than work-energy) say that the increased kinetic energy comes from the ______ of the _______ energy. Choose the best answer to fill in the blanks above: ANSWER: work / potential force / kinetic change / potential Correct Part E This process happens in such a way that total mechanical energy, equal to the ______ of the kinetic and potential energies, is _______. Choose the best answer to fill in the blanks above: ANSWER: sum / conserved sum / zero sum / not conserved difference / conserved Correct Orbiting Satellite A satellite of mass mm is in a circular orbit of radius R 2R_2 around a spherical planet of radius R 1R_1 made of a material with density ρrho. ( R 2R_2 is measured from the center of the planet, not its surface.) Use GG for the universal gravitational constant. Part A Find the kinetic energy of this satellite, KK. Express the satellite's kinetic energy in terms of GG, mm, πpi, R 1R_1, R 2R_2, and ρrho. Hint 1. Kinematics of circular motion The circular trajectory of the satellite implies that there is a certain inward radial acceleration a radial = v 2 / R 2, which must be due to the gravitational force F gF_g. Hint 2. Mass of the planet Find the mass of the planet in terms of its size and density. Express MM in terms of ρrho and R 1R_1. ANSWER: MM = 4πR 1 3ρ 3 Hint 3. Gravitational force Find F gF_g, the magnitude of the gravitational force on the satellite. Express your answer in terms of GG, mm, πpi, R 1R_1, R 2R_2, and ρrho. ANSWER: F gF_g = Gm 4πR 1 3ρ R22 3 Hint 4. Speed of the satellite Find the speed vv of the satellite. Express your answer in terms of GG, mm, πpi, R 1R_1, R 2R_2, and ρrho. ANSWER: vv = √ G 4πR 1 3ρ 3R 2 ANSWER: KK = 2GπR 1 3ρm 3R 2 Correct Part B Find UU, the gravitational potential energy of the satellite. Take the gravitational potential energy to be zero for an object infinitely far away from the planet. Express the satellite's gravitational potential energy in terms of GG, mm, πpi, R 1R_1, R 2R_2, and ρrho. Hint 1. What physical principle to use The gravitational potential energy associated with a 1 R2 force is best remembered. If you have to work it out, remember that the potential energy of an object found at height h = B, relative to height h = A, is equal to the negative of the work done by the gravitational force when the object is brought from AA to BB: U(B) = −W A → B, where B W A → B =∫ AF g dh. In this case, F gF_g is a function of the radius of the satellite trajectory, RR, and the zero point of the energy is conventionally taken at R =∞, so you want to use W ∞→R =∫ 2 R2 ∞ F g(R) dR. Hint 2. How to handle the math You will need to solve an integral of the form ∫ (C / x 2) dx, where CC is a constant. This evaluates to − C / x, and you will need to evaluate this between the limits of integration. ANSWER: UU = − 4GπR 1 3ρm Correct 3R 2 Part C What is the ratio of the kinetic energy of this satellite to its potential energy? K Express U in terms of parameters given in the introduction. ANSWER: K U = -0.500 Correct The result of this problem may be expressed as K U = n +1 2 where n = −2 is the exponent of the force law (i.e. F(R) = −G M m R n). This is a specical case of a general and powerful theroem of advanced classical mechanics known as the Virial Theorem. The theorem applies to the average of the kinetic and potential energies of any one or multiple objects moving over any closed (or almost closed) path that returns very close to itself provided that all objects interact via potentials with the same power law dependence on their separation. Thus it applies to stars in a galaxy, or masses tied together with springs (where F(R) = −k K U =1 since the force law is R 1). A Car with Constant Power The engine in an imaginary sports car can provide constant power to the wheels over a range of speeds from 0 to 70 miles per hour (mph). At full power, the car can accelerate from zero to 32.0 mph in time 1.40 s . Part A At full power, how long would it take for the car to accelerate from 0 to 64.0 mph ? Neglect friction and air resistance. Express your answer in seconds. Hint 1. Energy and power In the absence of friction, the constant power of the engine implies that the kinetic energy of the car increases linearly with time. Hint 2. Find the ratio of kinetic energies Find the (numerical) ratio of the car's kinetic energy K 1K_1 at speed 64.0 mph to K 2K_2, the kinetic energy at speed 32.0 mph . ANSWER: K1 K2 = 4 ANSWER: 5.60 s Correct Of course, neglecting friction, especially air friction, is completely unrealistic at such speeds. Part B A more realistic car would cause the wheels to spin in a manner that would result in the ground pushing it forward with a constant force (in contrast to the constant power in Part A). If such a sports car went from zero to 32.0 mph in time 1.40 s , how long would it take to go from zero to 64.0 mph ? Express your answer numerically, in seconds. Hint 1. How to approach the problem Constant force means constant acceleration. Use this fact to find how the speed increases with time. ANSWER: 2.80 s Correct This is probably the first and last time you will come across an imaginary car that goes slower than the real one! Potential Energy Graphs and Motion Learning Goal: To be able to interpret potential energy diagrams and predict the corresponding motion of a particle. Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle's kinetic energy, etc. Consider the potential energy diagram shown. The curve represents the value of potential energy UU as a function of the particle's coordinate xx. The horizontal line above the curve represents the constant value of the total energy of the particle EE. The total energy EE is the sum of kinetic (KK) and potential (UU) energies of the particle. The key idea in interpreting the graph can be expressed in the equation F x(x)= − dU( x ) dx , where F x(x) is the x component of the net force as function of the particle's coordinate xx. Note the negative sign: It means that the x component of the net force is negative when the derivative is positive and vice versa. For instance, if the particle is moving to the right, and its potential energy is increasing, the net force would be pulling the particle to the left. If you are still having trouble visualizing this, consider the following: If a massive particle is increasing its gravitational potential energy (that is, moving upward), the force of gravity is pulling in the opposite direction (that is, downward). If the x component of the net force is zero, the particle is said to be in equilibrium. There are two kinds of equilibrium: Stable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point (think of a ball rolling between two hills). Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill). In answering the following questions, we will assume that there is a single varying force FF acting on the particle along the x axis. Therefore, we will use the term force instead of the cumbersome x component of the net force. Part A The force acting on the particle at point A is __________. Hint 1. Sign of the derivative If a function increases (as xx increases) in a certain region, then the derivative of the function in that region is positive. Hint 2. Sign of the component If xx increases to the right, as in the graph shown, then a (one-dimensional) vector with a positive x component points to the right, and vice versa. ANSWER: directed to the right directed to the left equal to zero Correct Consider the graph in the region of point A. If the particle is moving to the right, it would be "climbing the hill," and the force would "pull it down," that is, pull the particle back to the left. Another, more abstract way of thinking about this is to say that the slope of the graph at point A is positive; therefore, the direction of → FF_vec is negative. Part B The force acting on the particle at point C is __________. Hint 1. Sign of the derivative If a function increases (as xx increases) in a certain region, then the derivative of the function in that region is positive, and vice versa. Hint 2. Sign of the component If xx increases to the right, as in the graph shown, then a (one-dimensional) vector with a positive x component points to the right, and vice versa. ANSWER: directed to the right directed to the left equal to zero Correct Part C The force acting on the particle at point B is __________. Hint 1. Derivative of a function at a local maximum At a local maximum, the derivative of a function is equal to zero. ANSWER: directed to the right directed to the left equal to zero Correct The slope of the graph is zero; therefore, the derivative dU / dx=0, and | Part D The acceleration of the particle at point B is __________. Hint 1. Relation between acceleration and force The relation between acceleration and force is given by Newton's 2nd law, F = ma. ANSWER: directed to the right directed to the left equal to zero → F | =0. Correct If the net force is zero, so is the acceleration. The particle is said to be in a state of equilibrium. Part E If the particle is located slightly to the left of point B, its acceleration is __________. Hint 1. The force on such a particle To the left of B, U(x)U(x) is an increasing function and so its derivative is positive. This implies that the x component of the force on a particle at this location is negative, or that the force is directed to the left, just like at A. What can you say now about the acceleration? ANSWER: directed to the right directed to the left equal to zero Correct Part F If the particle is located slightly to the right of point B, its acceleration is __________. Hint 1. The force on such a particle To the right of B, U(x)U(x) is a decreasing function and so its derivative is negative. This implies that the x component of the force on a particle at this location is positive, or that the force is directed to the right, just like at C. What can you now say about the acceleration? ANSWER: directed to the right directed to the left equal to zero Correct As you can see, small deviations from equilibrium at point B cause a force that accelerates the particle further away; hence the particle is in unstable equilibrium. Part G Name all labeled points on the graph corresponding to unstable equilibrium. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as . Hint 1. Definition of unstable equilibrium Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill). ANSWER: BF Correct Part H Name all labeled points on the graph corresponding to stable equilibrium. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as . Hint 1. Definition of stable equilibrium Stable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point. (Think of a ball rolling between two hills.) ANSWER: DH Correct Part I Name all labeled points on the graph where the acceleration of the particle is zero. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as . Hint 1. Relation between acceleration and force The relation between acceleration and force is given by Newton's 2nd law, F = ma. ANSWER: BDFH Correct Your answer, of course, includes the locations of both stable and unstable equilibrium. Part J Name all labeled points such that when a particle is released from rest there, it would accelerate to the left. List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as . Hint 1. Determine the sign of the x component of force If the acceleration is to the left, so is the force. This means that the x component of the force is __________. ANSWER: positive negative Hint 2. What is the behavior of U(x)U(x)? If the x component of the force at a point is negative, then the derivative of U(x)U(x) at that point is positive. This means that in the region around the point U(x)U(x) is __________. ANSWER: increasing decreasing ANSWER: AE Correct Part K Consider points A, E, and G. Of these three points, which one corresponds to the greatest magnitude of acceleration of the particle? Hint 1. Acceleration and force The greatest acceleration corresponds to the greatest magnitude of the net force, represented on the graph by the magnitude of the slope. ANSWER: A E G Correct Kinetic energy If the total energy EE of the particle is known, one can also use the graph of U(x)U(x) to draw conclusions about the kinetic energy of the particle since K = E − U. As a reminder, on this graph, the total energy EE is shown by the horizontal line. Part L What point on the graph corresponds to the maximum kinetic energy of the moving particle? Hint 1. KK, UU, and EE Since the total energy does not change, the maximum kinetic energy corresponds to the minimum potential energy. ANSWER: D Correct It makes sense that the kinetic energy of the particle is maximum at one of the (force) equilibrium points. For example, think of a pendulum (which has only one force equilibrium point--at the very bottom). Part M At what point on the graph does the particle have the lowest speed? ANSWER: B Correct As you can see, many different conclusions can be made about the particle's motion merely by looking at the graph. It is helpful to understand the character of motion qualitatively before you attempt quantitative problems. This problem should prove useful in improving such an understanding. Score Summary: Your score on this assignment is 97.3%. You received 136.22 out of a possible total of 140 points.