Wednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 1 PHYS 1443 – Section 001 Lecture #10 Wednesday, June 22, 2011 Dr. Jaehoon Yu • Potential Energy and the Conservative Force – Gravitational Potential Energy – Elastic Potential Energy • Conservation of Energy • Energy Diagram • General Energy Conservation & Mass Equivalence • More on gravitational potential energy • Escape speed • PowerWednesday, June 22, 2011 2 Special Project • Derive the formula for the gravitational acceleration ( ) at the radius from the center, inside of the Earth. (10 points) • Compute the fractional magnitude of the gravitational acceleration 1km and 500km inside the surface of the Earth with respect to that on the surface. (6 points, 3 points each) • Due at the beginning of the class Monday, June 27 PHYS 1443-001, Spring 2011 Dr. Jaehoon YuWednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 3 Valid Planetarium Shows • Regular shows – TX star gazing; Nanocam; Ice Worlds • Private shows for a group of 15 or more – Bad Astronomy; Black Holes; IBEX; Magnificent Sun – Microcosm; Stars of the Pharaohs; Time Space – Two Small Pieces of Glass; SOFIA – Violent Universe; Wonders of the Universe • Please watch the show and obtain the signature on the back of the ticket stubWednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 4 Potential Energy & Conservation of Mechanical Energy Energy associated with a system of objects Stored energy which has the potential or the possibility to work or to convert to kinetic energy What does this mean? In order to describe potential energy, U, a system must be defined. What are other forms of energies in the universe? The concept of potential energy can only be used under the special class of forces called the conservative force which results in the principle of conservation of mechanical energy. Mechanical Energy Biological Energy Electromagnetic Energy Nuclear Energy Chemical Energy These different types of energies are stored in the universe in many different forms!!! If one takes into account ALL forms of energy, the total energy in the entire universe is conserved. It just transforms from one form to another.Wednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 5 Gravitational Potential Energy When an object is falling, the gravitational force, Mg, performs the work on the object, increasing the object’s kinetic energy. So the potential energy of an object at height h, the potential to do work, is expressed as This potential energy is given to an object by the gravitational field in the system of Earth by virtue of the object’s height from an arbitrary zero level m hf m mg hi What does this mean? The work done on the object by the gravitational force as the brick drops from hi to hf is: Work by the gravitational force as the brick drops from yi to yf is the negative change of the system’s potential energy Potential energy was spent in order for the gravitational force to increase the brick’s kinetic energy. ΔPE = PEf− PEi(since )Wednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 6 The gymnast leaves the trampoline at an initial height of 1.20 m and reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast? Ex. A Gymnast on a TrampolineWednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 7 Ex. Continued From the work-kinetic energy theorem Work done by the gravitational force Since at the maximum height, the final speed is 0. Using work-KE theorem, we obtainWednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 8 Conservative Forces and Potential Energy The work done on an object by a conservative force is equal to the decrease in the potential energy of the system What does this statement tell you? The work done by a conservative force is equal to the negative change of the potential energy associated with that force. We can rewrite the above equation in terms of the potential energy U So the potential energy associated with a conservative force at any given position becomes Only the changes in potential energy of a system is physically meaningful!! Potential energy function What can you tell from the potential energy function above? Since Ui is a constant, it only shifts the resulting Uf(x) by a constant amount. One can always change the initial potential so that Ui can be 0.Wednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 9 More Conservative and Non-conservative Forces When directly falls, the work done on the object by the gravitation force is The work done on an object by the gravitational force does not depend on the object’s path in the absence of a retardation force. How about if we lengthen the incline by a factor of 2, keeping the height the same?? Still the same amount of work Forces like gravitational and elastic forces are called the conservative force So the work done by the gravitational force on an object is independent of the path of the object’s movements. It only depends on the difference of the object’s initial and final position in the direction of the force. h l m θ"mg When sliding down the hill of length l, the work is N Total mechanical energy is conserved!! 1. If the work performed by the force does not depend on the path. 2. If the work performed on a closed path is 0.Wednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 10 Example for Potential Energy A bowler drops bowling ball of mass 7kg on his toe. Choosing the floor level as y=0, estimate the total work done on the ball by the gravitational force as the ball falls on the toe. b) Perform the same calculation using the top of the bowler’s head as the origin. Assuming the bowler’s height is 1.8m, the ball’s original position is –1.3m, and the toe is at –1.77m. M Let’s assume the top of the toe is 0.03m from the floor and the hand was 0.5m above the floor. What has to change? First we must re-compute the positions of the ball in his hand and on his toe.Wednesday, June 22, 2011 PHYS 1443-001, Spring 2011 Dr. Jaehoon Yu 11 Elastic Potential Energy The force spring exerts on an object when it is distorted from its equilibrium by a distance x is Potential energy given to an object by a spring or an object with elasticity in the
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