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CHAPTER 8

CHAPTER 8. 8: Potential Energy & Conservation of Energy. Potential energy (U) is the energy which can be associated with configuration of a systems of objects. One example is GRAVITATIONAL POTENTIAL ENERGY , associated with the separation between two objects attracted to each

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CHAPTER 8

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  1. CHAPTER 8

  2. 8: Potential Energy & Conservation of Energy Potential energy(U) is the energy which can be associated with configuration of a systems of objects. One example is GRAVITATIONAL POTENTIAL ENERGY, associated with the separation between two objects attracted to each other by the gravitational force. By increasing the distance between two objects (e.g. by lifting an object higher) the work done on the gravitational force increases the gravitational potential energy of the system. Another example is ELASTIC POTENTIAL ENERGY which is associated with compression or extension of an elastic object (such as a perfect spring). By compressing or extending such a spring, work is done against the restoring force which in turn increases the elastic potential energy in the spring. REGAN PHY34210

  3. Work and Potential Energy In general, the change in potential energy, DU is equal to the negative of the work done (W) by the force on the object (e.g., gravitational force on a falling object or the restoring force on a block pushed by a perfect spring), i.e., DU=-W Conservative and Non-Conservative Forces If work, W1, is done, if the configuration by which the work is done is reversed, the force reverses the energy transfer, doing work, W2. If W1=-W2, whereby kinetic energy is always transferred to potential energy, the force is said to be a CONSERVATIVE FORCE. The net work done by a conservative force in a closed path is zero. The work done by a conservative force on a particle moving between 2 points does not depend on the path taken by the particle. NON-CONSERVATIVE FORCES include friction, which causes transfer from kinetic to thermal energy. This can not be transferred back (100%) to the original mechanical energy of the system. REGAN PHY34210

  4. Determining Potential Energy Values REGAN PHY34210

  5. Conservation of Mechanical Energy REGAN PHY34210

  6. The Potential Energy Curve In the general, the force at position x, can be calculated by differentiating the potential curve with respect to x (remembering the -ve sign). F(x) is minus the SLOPE of U(x) as a function of x REGAN PHY34210

  7. Turning Points For conservative forces, the mechanical energy of the system is conserved and given by, U(x) + K(x) = Emec where U(x) is the potential energy and K(x) is the kinetic energy. Therefore, K(x) = Emec-U(x). Since K(x) must be positive ( K=1/2mv2), the max. value of x which the particle has is at Emec=U(x) (i.e., when K(x)=0). Note since F(x) = - ( dU(x)/dx ) , the force is negative. Thus the particle is ‘pushed back. i.e., it turns around at a boundary. REGAN PHY34210

  8. Equilibrium Points Equilibrium Points:refer to points where, dU/dx=-F(x)=0. Neutral Equilibrium: is when a particle’s total mechanical energy is equal to its potential energy (i.e., kinetic energy equals zero). If no force acts on the particle, then dU/dx=0 (i.e. U(x) is constant) and the particle does not move. (For example, a marble on a flat table top.) Unstable Equilibrium: is a point where the kinetic energy is zero at precisely that point, but even a small displacement from this point will result in the particle being pushed further away (e.g., a ball at the very top of a hill or a marble on an upturned dish). Stable Equilibrium: is when the kinetic energy is zero, but any displacement results in a restoring force which pushes the particle back towards the stable equilibrium point. An example would be a marble at the bottom of a bowl, or a car at the bottom of a valley. REGAN PHY34210

  9. U(x) x D B C A Particles at A,B, C and D are in at equilibrium points where dU/dx = 0 A,C are both in stable equilibrium ( d 2U/dx2 = +ve ) B is an unstable equilibrium ( d 2U/dx2 = -ve ) D is a neutral equilibrium ( d 2U/dx2 = 0 ) REGAN PHY34210

  10. Work Done by an External Force Previously we have looked at the work done to/from an object. We can extend this to a system of more than one object. Work is the energy transferred to or from a system by means of an external force acting on that system. No friction (conservative forces) Including friction REGAN PHY34210

  11. Conservation of Energy This states that ‘ The total energy of a system, E, can only change by amounts of energy that are transferred to or from the system. ’ Work done can be considered as energy transfer, so we can write, If a system is ISOLATED from it surroundings, no energy can be transferred to or from it. Thus for an isolated system, the total energy of the system can not change, i.e., Another way of writing this is, which means that for an isolated system, the total energies can be related at different instants, WITHOUT CONSIDERING THE ENERGIES AT INTERMEDIATE TIMES. REGAN PHY34210

  12. Example 1: A child of mass m slides down a helter skelter of height, h. Assuming the slide is frictionless, what is the speed of the child at the bottom of the slide ? h=10m REGAN PHY34210

  13. Example 2: A man of mass, m, jumps from a ledge of height, h above the ground, attached by a bungee cord of length L. Assuming that the cord obeys Hooke’s law and has a spring constant, k, what is the general solution for the maximum extension, x, of the cord ? L h x m REGAN PHY34210

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