AP PHYSICS 1. Energy 2016 EDITION

Pre-Assessment Questions Consider a system which could consist of any of these three: a block, a spring, and the Earth. The block s kinetic energy is K, the block-earth potential energy is U g, and the spring s potential energy is U s. For the three situations shown below, describe in words a situation and draw a diagram in which the energy transformation shown takes place. P1. P2. P3. P4. Describe in words, draw a picture diagram, and draw an energy bar chart representing a situation that meets both of these criteria at the same time. The mechanical energy of the block-earth system increases, while at the same time, The mechanical energy of the block-spring-earth system decreases. Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 1

Energy is like the money of physics. Money gives people the ability to do things like buy groceries, take vacations, and receive services. Energy is what allows objects or systems to undergo changes, such as changing speed, changing temperature, changing height, etc. With few exceptions, any time an object or system undergoes any sort of change, it is because an energy transformation took place. An energy transformation could be: Energy transforming from one form to another (example: kinetic to potential energy) Energy being transferred from one object to another (example: bowling ball hits bowling pin) Energy being transferred from one system to another Energy being transferred from (or to) a system to (or from) its surroundings. Energy is measured in units called joules (unit symbol: J) and, like money, is a scalar quantity (no concept of direction is associated with these quantities). It is possible in some cases to have a negative amount of energy, which is similar to having a negative amount of money or debt. The types of energy are as follows: Kinetic Energy (KE) (algebraic symbol: K, basic equation: K = ½mv 2 ) is the energy that a moving object has simply because it moves. Kinetic energy can never be negative. Gravitational Potential Energy (GPE) (algebraic symbol: U g, basic equation: ΔU g = mgδy) is the energy that exists because an object (mass m) is separated (height y) from another object that creates gravitational field g. As the object gains height (becomes more separated from the Earth), the gravitational potential energy increases. Spring (or Elastic) Potential Energy (SPE) (algebraic symbol: U s, basic equation: U s = ½kx 2 ) is the energy that exists because a spring is stretched or compressed (a distance x) from its natural length. Potential energy increases if the spring is stretched or compressed farther from its natural length. Internal Energy (sometimes called Thermal Energy) is the energy that an object has because its molecules vibrate (which causes the object to have a temperature). The more the molecules vibrate, the greater the internal energy of the object, and the higher the object s temperature. Other forms of energy include sound energy (the energy of vibrating air molecules that carry sound waves), light energy (the energy carried by electromagnetic radiation), electrical energy (the energy carried by electric charges as they travel around a circuit), and chemical energy (energy stored in chemical bonds, the type of energy gasoline has before it is burned to make your car go). A force has the ability to transfer energy (from one object to another, system to another, or form to another). When a force transfers energy, the force does work. Work is any transfer of energy. Work is a quantity measured in joules (like energy) because the amount of work is the amount of energy that was transferred. If work is positive, then that is a gain in energy (getting more energy or an increase in energy). If work is negative, then that is a loss of energy (giving away energy or a decrease in energy). A force does work if the force acts on the object while the object moves from one point to another. The sign of work depends on the angle between the direction of the force and the direction of the displacement: Work is positive if the direction of force and direction of displacement are acute. Work is negative if the direction of force and direction of displacement are obtuse. No work is done by a force if the direction of force and direction of displacement are perpendicular. The equation for the work W done on an object by a force F while the object is displaced a distance d is W = Fd cosθ. The θ is the angle between the force s direction and the direction the object is displaced. The cosθ in the equation enforces the three bullet-point rule about sign and angle above. The full equation as it Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 2

appears on the AP exam equation chart is Δ E = W = F d = Fd cosθ. This is to remind you that work (W) is a change in energy (ΔE), and that you need not use cosθ if force and distance are parallel (F d). Suppose that you move from one point to another, and while you are moving, four forces act on you. Each force does some work, so there must be a total amount of work done by all the forces. We call this the net work (net means total of all the numbers, but some numbers can be negative ). The net amount of work done on an object or system is the net result of all energies given (positive work) and all energies taken away (negative work). The Work-Energy Theorem tells us what happens when there is a net amount of work on a system: Work-Energy Theorem (The net amount of work done on an object or system) = (The change in the object or system s kinetic energy) If positive net work is done on an object, the object gains KE and speeds up. If negative net work is done on an object, the object loses KE and slows down. If an object neither speeds up nor slows down, you can know for sure that the net work done on the object is zero (which means the object is gaining as much energy as it is losing). A term that comes up a lot on the AP Physics exam is mechanical energy (ME). Mechanical energy is kinetic energy plus potential energy. Mechanical energy does not count internal/thermal, light, sound, or chemical energy. Mechanical energy is usually measured for a system, and adds up as follows: If the system includes an object that can move, then that object s kinetic energy adds to the system ME. If the system includes an object that exerts gravity (like Earth) and an object that is pulled on by gravity (like anything), then gravitational potential energy adds to the system ME. If the system includes a spring or any other elastic object, then elastic potential energy adds to the ME. Example: The block compresses the spring, and there is no friction anywhere. The block is released at A. The block separates from the spring at B. The block becomes as projectile at C. The block is just about to strike the ground at D. System Consisting of Mechanical Energy is Between A & B, mech. energy Between B & C, mech. energy Between C & D, mech. energy Block only KE Increases Remains Constant Increases Block & Earth KE + GPE Increases Remains Constant Remains Constant Block & Spring KE + SPE Remains Constant Remains Constant Increases Block-Earth-Spring KE + GPE + SPE Remains Constant Remains Constant Remains Constant If a system does not receive energy from or give energy to its surroundings, then the system obeys the Law of Conservation of Energy (which says that the total energy of a system before equals the total energy of the system after ). If there are no forces (like friction) that convert KE and PE into other forms of energy, then we can apply the more strict Conservation of Mechanical Energy (which is true if all of the energy transformations are KE to PE and PE to KE). Conservation of Mechanical Energy looks like this equation: K initial + U initial = K final + U final Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 3

Power tells us how quickly energy is converted from one form to another. Power (algebraic symbol P) is measured in units called watts (unit symbol: W). The number of watts is the number of joules of energy transferred every second. Remembering that work done is the same as amount of energy transferred: ΔE amount of energy transferred W amount of work done P = = = = Δt time it took to transfer Δt time it took to do Multiple-Choice Questions M1. Cars X and Y travel on a straight, level roadway each at constant but different speeds. Car X is light and travels fast; car Y is heavy and travels slow. The driver of each car applies the brakes for a short interval so that both cars experience the same net braking force F for the same interval of time Δt. Neither car comes to a complete stop. Which of the following statements is true regarding the loss of kinetic energy by each car as a result of the braking procedure? (A) Car X loses more kinetic energy because of its high speed. (B) Car Y loses more kinetic energy because of its higher mass. (C) The same net force acts on both cars, so both cars lose the same kinetic energy. (D) The braking force acts on both cars for the same time, so both cars lose the same kinetic energy. M2. The diagram shows a block set on a frictionless table. The block is connected to a hanging bucket by a string that passes over an ideal pulley. When the bucket is released, the bucket accelerates toward the floor. Let E M-E represent the mechanical energy of the block-earth system, and E M-b-E represent the mechanical energy of the block-bucket-earth system. Which pair of graphs shows how these energies change as functions of time from the moment the bucket is released? (A) (C) (B) (D) M3. A car travels along a road that lies on the side of a hill. In which of the following cases does the mechanical energy of the car-earth system increase? (A) The car travels uphill with constant speed. (B) The car travels downhill with constant speed. (C) The car travels uphill only under the influence of M4. A 2 kg mass is released from the top of an 8 m Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 4

gravity and the normal force. (D) The car travels downhill only under the influence of gravity and the normal force. high incline. Upon reaching the bottom, the mass is observed to move with a speed of 10 m/s. How much work does the force of friction do on the mass as it slides down the incline? (A) 60 J (B) 100 J (C) 160 J (D) 200 J M5. A force F applied to a object of mass m causes it to move in a straight line a distance D during an interval of time T. The object gains kinetic energy K during this interval. In which of the following cases will the object gain the same kinetic energy K? (A) A force F is applied to an object of mass 2m during a time interval T. (B) A force F is applied to an object of mass 2m as it travels a distance D. (C) A force 2F is applied to an object of mass 2m during a time interval T. (D) A force 2F is applied to an object of mass 2m as it travels a distance D. M6. A block slides on a frictionless track that has a spring attached to one end. The potential energy U and kinetic energy K of the block-earth-spring system are shown in the graph as functions of time. Which of the following could describe the interaction that the block undergoes that would produce this graph? Select two answers. (A) The block slides down one track and up another track. (B) The block slides along a horizontal track, up an angled track, and back down the same angled track. (C) The block slides along a horizontal track, compresses a spring, and is launched back by the spring. (D) The block is attached to a spring and wiggles back and forth. Free-Response Questions F1. A ball of mass m is connected to a string of length L. The string is not very strong and there is a risk that the string will break if the string is put under too much tension. A student raises the ball so that the string is initially horizontal and releases the ball from rest. The ball swings through the vertical position without the string breaking. (a) The dot below represents the ball at the instant it is in the vertical position. On the dot, draw and label the forces (NOT components) acting on the ball at this instant. The length of each vector should represent the relative magnitude of the force it represents. Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 5

(b) Write the following equations in terms of m, L, g, T (the tension in the string), and v (the speed of the ball). i. A statement of Newton s Second Law relating the forces you drew in part (a) ii. A statement of Conservation of Mechanical Energy between the moment the ball is released and the moment the ball passes through the vertical position (c) Suppose the experiment is repeated with the same type of string, but the length of the string is different. Student A is concerned that the string will break if the string is made longer, since that would result in the bowling ball moving faster at the bottom of its swing. Student B is concerned that the string will break if the string is made shorter, since a smaller circle results in greater acceleration. i. Which student is correct, or is neither student correct? Student A Student B Neither ii. Using the equations you wrote in part (b), justify your answer. Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 6

F2. The diagram shows a new toy that consists of a car of mass m that rolls on nearly-frictionless bearings and a spring-loaded launcher. An apparatus (not shown) within the launcher allows a person to compress the spring any distance, and then release the spring so that the spring transfers all of its energy to the car. A student is given the toy and asked to make a hypothesis about the relationship between the spring s compression and the subsequent motion of the car. (a) The student hypothesizes that, if the cart is launched horizontally, the launch speed v of the car is directly proportional to the compression distance x of the spring. i. Write an equation that relates v and x. The equation will necessarily contain some constant quantities. Use the equation to evaluate the student s hypothesis. ii. Outline a procedure whereby the student could use commonly available equipment to make measurements in order to test the hypothesis. Draw and label a diagram of the experimental setup. iii. Explain the process of data analysis. How will the measurements be used to evaluate the hypothesis? (b) The student glues a piece of felt to the top of the car, and then orients the car upside down within the launcher so that the car is launched and slides to rest on the felt due to the effects of a constant friction force. The student hypothesizes that the sliding distance D of the car is directly proportional to the compression distance x of the spring. i. Write an equation that relates v and D. The equation will necessarily contain some constant quantities. Use the equation to evaluate the student s hypothesis. Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 7

ii. After performing several trials, the student creates the data table shown below. Do the data support the student s hypothesis? Explain your reasoning. Compression Distance x (m) 0.01 0.02 0.03 0.04 0.05 0.06 Sliding Distance D (m) 0.051 0.204 0.459 0.816 1.276 1.837 F3. A spring of force constant k is set into a hole in a table and a light platform is attached to the top. The spring extends a distance D out of the table s surface. A mass m is set on the platform and the spring is compressed so that the platform is initially at the same height as the table s surface. When the mass is released, the mass is projected directly upward and reaches a maximum height of 4D. (a) Write an expression for D in terms of m, g, and k. (b) From the instant that the mass is released until the mass reaches its highest height, the mechanical energy of the mass-earth-spring system is E. This is taking the table s surface as the zero-point for gravitational potential energy and the spring s natural length as the zero-point for elastic potential energy. Sketch graphs of the following quantities as functions of the mass s height above the table s surface as it is projected. i. U g, the gravitational potential energy of the mass-earth system ii. U s, the elastic potential energy of the spring iii. K, the kinetic energy of the mass Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 8

(c) The spring is re-set in the hole so that a length 3D of the spring extends above the table. A new mass 3m is set on the spring s platform, and the spring is again compressed until the platform is at the same level as the table. Before the mass is released, a student observing the situation makes the following prediction: The spring will have three times more energy stored in it, but the mass is also three times as much. It takes three times as much energy to raise triple the mass to the same height, so the mass in this new situation should reach the same maximum height that the other mass reached before. i. Which aspects of the student s statement are correct? Cite specific physical principles or relationships to explain your reasoning. ii. Correct any incorrect aspects of the student s statement. Cite specific physical principles or relationships to explain your reasoning. iii. Explain whether the mass reaches the same height, higher, or lower than in the original case. Explain your reasoning using appropriate principles and relationships. F4. Two identical stunt professionals A and B stand on the roof of building 1. Person A steps off of the roof and falls vertically onto the solid ground. At the same instant, person B steps off the roof while holding onto a rope with its opposite end is fixed to building 2 of the same height. The rope is initially horizontal, and after swinging downward almost the same vertical distance as A fell, person B collides Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 9

with the solid wall of building 2. (a) Which person will sustain more serious injuries, or will both people sustain injuries of the same seriousness? Explain your reasoning. (b) Which person will reach his impact point in less time, or will both people reach their impact points at the same time? Explain your reasoning in a coherent paragraph-length response that discusses appropriate physical principles. Copyright 2016 National Math + Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 10