LECTURE 14 CONSERVATION OF ENERGY Instructor: Kazumi Tolich Lecture 14 2 ¨ Reading chapter 8-4 to 8-5 ¤ Conservation of energy ¤ Potential energy curves and equipotentials Quiz: 1 3 ¨ A student wants to lift a 50-kg mass, so she hooks it up to the pulley system as shown. The resulting force required to lift the mass is now half the weight of the mass. If the mass rises by 10 cm, how much work did the student do pulling the rope at a constant speed in Joules? Mass Quiz: 14-1 answer 4 ¨ 50 J ¨ If the mass is raised by 10 cm, then both of the right ropes are now 10 cm shorter. The total rope length must be the same, so the left rope is 20 cm longer. So the student pulled it by 20 cm. ( *+ ,- /0 1.34 5⁄67 ¨ Work = �� = � = � = 0.20 m = 50 J ) ) ) ¨ This is equal to the change in mechanical energy of the Earth-mass system. ) ¨ ∆� = ∆� + ∆� = ��∆� = 50 kg 9.81 m⁄s 0.10 m = 50 J Mass Quiz: 2 5 ¨ Suppose a book is sliding on a horizontal tabletop, slowing down due to the friction between the book and the table surface. Eventually the book stops. Define a system containing the book and the table. Which of the following statements is/are correct during this process? Choose all that apply. A. The mechanical energy of the system is conserved. B. The total energy of the system is conserved. Quiz: 14-2 answer 6 ¨ The total energy of the system is conserved. 4 ) ¨ Initially, the book is moving, so the system has a non-zero kinetic energy. � = �� ) ¨ Finally, the book stops on top of the stationary table, so the system does not have kinetic energy. ¨ The kinetic energy decreased to zero, so the mechanical energy is not conserved: ¨ ∆� = ∆� + ∆� ≠ 0 ¨ The reduced mechanical energy went to heat up the table and book surfaces through the friction. ¨ The energy associated with the temperature of a system is its internal energy. Change in mechanical energy 7 ¨ If in a system an object slides through a distance � with a kinetic friction �/ applied on it, the mechanical energy of the system changes. ∆� = −�/� ¨ Other factors that changes mechanical energy: ¤ Air or water resistance ¤ Energy to deform an object result in increase of thermal energy. ¤ If chemical reaction takes place within the system, change in chemical energy occurs. ¤ Other processes can change nuclear, electromagnetic, or other forms of energy. Quiz: 3 ¨ A body falls through the atmosphere gaining 20J of kinetic energy. Air resistance is not negligible. If you define a system containing the body, its surrounding air, and the earth, how much gravitational potential energy did the system lose? A. 20 J B. more than 20 J C. less than 20 J Quiz: 14-3 answer ¨ more than 20 J ¨ As the body falls, the gravitational potential energy is converted to kinetic energy of the body as well as internal energy of the air and the body as they heat up. ¨ −∆� = ∆� + ∆�OPQ = 20 J + ∆�OPQ Example: 1 10 ¨ At a playground, a child of mass � = 18 kg plays on a slide that drops through a height of ℎ = 2.2 m. The child starts at rest at the top of the slide. On the way down, the friction causes the slide and child to heat up, creating ∆�OPQ = 373 J of internal energy. What is the child’s speed at the bottom of the slide? Law of conservation of energy/Demo: 1 11 ¨ The law of conservation of energy: Within the isolated system, energy can be converted from one form to another, or transmitted from one region to another, but energy can never be created or destroyed. The change in the total energy of a system is equal to the energy that enters minus the energy that leaves the system. ¨ The law of conservation of energy is one of the most fundamental laws of physics. We have never observed any violation of this law. ¨ Demo: bowling ball pendulum “Conserving” energy 12 ¨ If energy cannot be created or destroyed, why do we bother “conserving” energy by turning off unused lights, etc? ¤ There are useful forms of energy and not so useful form (thermal energy). Converting thermal energy to more useful form of energy does not have 100% efficiency. ¨ On “Earth Day,” many people tried to “conserve” electric energy by turning off the lights and lighting candles. What is wrong with this picture? Example: 2 13 ¨ A moving block with a mass � = 2.5 kg collides with a horizontal spring whose spring constant is � = 320 N/m. The block compresses the spring a maximum distance of 7.5 cm from its rest position. The coefficient of kinetic friction between the block and the horizontal surface is �/ = 0.25. a) How much work is done by the spring in bringing the block to rest? b) How much mechanical energy is dissipated by the force of friction while the block is being brought to rest by the spring? c) What is the speed of the block when it hits the spring? Gravitational potential energy curves 14 ¨ A ball on a frictionless track starts at rest at A. ¨ The ball-Earth system has the least potential energy, and the most kinetic energy at B. ¨ The ball turns around at D (turning point) and repeats the process. 2D gravitational potential energy plot 15 ¨ Lines corresponding to constant values of potential energy are called equipotentials. ¨ Since � = ���, where � is the height above where you define � to be zero, equipotential lines for gravitational potential energy is at the same height. Elastic potential energy curves 16 ¨ A mass attached to a spring is pulled by distance � from the equilibrium position and released. ¨ The mass undergoes an oscillatory motion. 1 � = ��) 2 Quiz: 4 17 ¨ The graph shows the gravitational potential energy of an object-Earth system as a function of position. When the object is moving in the negative �-direction at � = 0.15 m, the kinetic energy is � = 30 J. What value of � does the object change its direction of motion? Indicate the position on the �-axis. Quiz: 13-4 answer 18 ¨ The total mechanical energy is ~650 J. � = 30 J � � Turning point.