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PHYS 1443 – Section 001 Lecture #10

PHYS 1443 – Section 001 Lecture #10. Thursday June 16, 2005 Dr. Andrew Brandt. Energy Diagram and Equilibrium Gravitational Potential Energy Power Test. Announcements. Test today at 8:50 I plan to post interim grades on Web Friday afternoon Homework: HW6 on ch 7 due Monday 6/20 at 8pm

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PHYS 1443 – Section 001 Lecture #10

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  1. PHYS 1443 – Section 001Lecture #10 Thursday June 16, 2005 Dr. Andrew Brandt • Energy Diagram and Equilibrium • Gravitational Potential Energy • Power • Test PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt

  2. Announcements • Test today at 8:50 • I plan to post interim grades on Web Friday afternoon • Homework: • HW6 on ch 7 due Monday 6/20 at 8pm • HW7 on ch 8 due Tuesday 6/21 at 8pm PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt

  3. How are Conservative Forces Related to Potential Energy? Work done by a force component on an object through a displacement Dx is For an infinitesimal displacement Dx Results in the conservative force-potential relationship This relationship says that any conservative force acting on an object within a given system is the same as the negative derivative of the potential energy of the system with respect to position. 1. spring-ball system: Does this statement make sense? 2. Earth-ball system: The relationship works in both the conservative force cases we have studied! PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt

  4. Us Minimum Stable equilibrium x -xm xm Maximum unstable equilibrium Energy Diagram and the Equilibrium of a System One can draw potential energy as a function of position  Energy Diagram Let’s consider potential energy of a spring-ball system What shape is this diagram? A Parabola What does this energy diagram tell you? • Potential energy for this system is the same independent of the sign of the position. • The force is 0 when the slope of the potential energy curve is 0 at a position. • x=0 is a stable equilibrium point of this system. Position of a stable equilibrium corresponds to a point where the potential energy is at a minimum. Position of an unstable equilibrium corresponds to a point where the potential energy is a maximum. PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt

  5. General Energy Conservation and Mass-Energy Equivalence General Principle of Energy Conservation The total energy of an isolated system is conserved as long as all forms of energy are taken into account. Friction is a non-conservative force and causes mechanical energy to change to other forms of energy. What about friction? However, if you add the new forms of energy altogether, the system as a whole did not lose any energy, as long as it is self-contained or isolated. In the grand scale of the universe, no energy can be destroyed or created but just transformed or transferred from one place to another. Total energy of universe is constant!! In any physical or chemical process, mass is neither created nor destroyed. Mass before a process is identical to the mass after the process. Principle of Conservation of Mass Einstein’s Mass-Energy equality. How many joules does your body correspond to? PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt

  6. Where is the unit vector pointing outward from the center of the Earth E The Gravitational Field The gravitational force is a field force. The force exists everywhere in the universe. If one were to place a test object of mass m at any point in space in the existence of another object of mass M, the test object will feel the gravitational force exerted by M, Therefore the gravitational field g is defined as In other words, the gravitational field at a point in the space is the gravitational force experienced by a test particle placed at the point divided by the mass of the test particle. So how does the Earth’s gravitational field look? Far away from the Earth’s surface Close to the Earth’s surface PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt

  7. The Gravitational Potential Energy What is the potential energy of an object at a height y from the surface of the Earth? Do you think this relation is generally true? No Why not? Because this formula is only valid for the case where the gravitational force is constant, near the surface of the Earth and the generalized gravitational force is inversely proportional to the square of the distance. OK. Then how would we generalize the potential energy in the gravitational field? PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt

  8. More on the Gravitational Potential Energy Since the gravitational force is a radial force, it performs work only when the path is in the radial direction. Therefore, the work performed by the position dependent gravitational force becomes For the whole path Potential energy is the negative of the change of work along the path The Earth’s gravitational force is So the potential energy function becomes Since only the difference in potential energy matters, by taking the infinite distance as the initial point of the potential energy, we obtain For any two particles? The energy needed to take the particles infinitely apart. For many particles? PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt

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