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Chapter 8: Potential Energy and Conservative Forces • Potential Energy and Conservation of Energy. • Conservative and non-conservative forces • Gravitational and Elastic Potential Energy • Conservation of (Mechanical) Energy • External and Internal Forces • CONSERVATION OF ENERGY 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 1/8 8. 2 Work and Potential Energy Potential Energy U is a form of stored energy that can be associated with the configuration (or arrangement) of a system of objects that exert certain types of forces (conservative) on one another. - When work gets done on an object, its potential and/or kinetic energy increases. - There are different types of potential energy: 1. Gravitational energy 2. Elastic potential energy (energy in an stretched spring) 3. Others (magnetic, electric, chemical, …) 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 2/8 • We know that the work done result in a change in kinetic energy (W=DK). Now we can ask the question: where did the kinetic energy go (if it is decreased) or where did it come from (if it increased)! Note that the force only function as the agent which rearranges the configuration of the system (by displacing one or more of the object in the system). Assuming that our system is isolated (no external force acting on it) the answer, as you have already guessed, is to (or from) the potential energy of the system.. • When one of these special forces (let us label it Fc) does some work (Wc) by changing the system configuration, the force derives the energy from the stored potential energy associated with that force: U decreases DUc Wc WW 00 U increases 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 3/8 Conservative and Nonconservative forces • We can define a potential energy for this force by the equation DU=-W only if W12= - W21. • A force for which W12= - W21 is called a conservative forces. This is same as saying that the net work done by a conservative force around any closed path (return back to the initial configuration) is zero. A force that is not conservative is called a nonconservative force. We cannot define potential energy associated with a nonconservative forces. 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 4/8 • The gravitational force and the spring force are examples of conservative forces. The frictional force and fluid drag force are examples of nonconservative forces. path inependance of conservative forces The work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle. Wab,1 Wab,2 The net work done by a conservative force on a particle moving moving around every closed is zero. Wab,1 Wba ,2 0 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5/8 Non-conservative forces: A force is non-conservative if it causes a change in mechanical energy; mechanical energy is the sum of kinetic and potential energy. Example: Frictional force. - This energy cannot be converted back into other forms of energy (irreversible). - Work does depend on path. Sliding a book on a table 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 6/8 8.4 Determining Potential Energy Values • In general we can write: x y z DU W Fx dx Fy dy Fz dz f x y i • f f z i i Gravitational potential energy, DUg y y DUg Wg Fg dy (mg ) dy f y f y i i mg y Fg has only y-component: (-mg ) ˆj yf yi DUg mg ( y y ) mg Dy f i Rewrite Uf as U and yf as y and take Ui to be the reference level and assign the values Ui =0 and yi =0. U ( y) mg y DO CP 8-2 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 7/8 • Elastic potential energy, DUs xi xf 1 2 2 DU s W s k (x f x i ) 2 Elastic potential energy stored in a spring: 1 2 U s kx 2 5-Nov-06 The spring is stretched or compresses from its equilibrium position by x Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 8/8 SP 8-2 • • 5-Nov-06 (a) What is the gravitational potential energy U of the sloth–Earth system if we take the reference point y = 0 to be (1) at the ground, (2) at a balcony floor that is 3.0 m above the ground, (3) at the limb, and (4) 1.0 m above the limb? Take the gravitational potential energy to be zero at y = 0. (b) The sloth drops to the ground. For each choice of reference point, what is the change DU in the potential energy of the sloth-Earth system due to the fall? Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 9/8 8.5 Conservation of Mechanical Energy If we deal only with conservative forces and If we deal with an isolated system (no energy added or removed): The total mechanical energy Emec K U of a system remains constant!!!! DEmec DK DU 0 K2 U 2 K1 U1 The final and initial energy of a system remain the same: Thus: 5-Nov-06 Ei = Ef E Ki U i K f U f Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 10/8 • 5-Nov-06 Figure 8-7 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 11/8 8-7 Work done on a System by an External Force (and friction) • When we stated the conservation of mechanical energy for a system in the previous section, we specified two conditions: • • • Isolated system (no external forces) Only conservative forces in the system. Let us now introduce external forces doing work on the system, then: Wex DK DU Wex DEmec • And also add nonconservative forces (friction involved) in the system: Wex DEmec DEth Eth f k d k N d 5-Nov-06 ( work done on the system, no friction invloved ) (work done on the system, friction involved) (increase in thermal energy by sliding) Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 12/8 8-8 Conservation of Energy • The total energy of a system can change only by amounts of energy Wex that are transferred to or from the system. Wex DEmec DEth DEint • DEint acknowledges the fact that thermal energy is not the only other form of energy that a system can have which is not mechanical energy, e.g. chemical energy in your muscles or in a battery, or nuclear energy. • The total energy of an isolated system cannot change. DEmec DEth DEint 0 • Power as the rate at which energy is transferred from one form to another Pavg DE Dt P dE dt Average power Instantaneous power The rate at which the work is done is a special case of energy being transferred to (or from) kinetic energy (one form of energy). 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 13/8 P.5 A frictionless roller coaster of mass m 825 kg tops the first hill with v0 = 17.0 m/s, at the initial height h = 42.0 m. How much work does the gravitational force do on the car from that point to (a) point A ? (b) point B, and (c) point C? If the gravit. Pot. Energy of the car-Earth system is taken to be zero at C, what is its value when the car iis at (d) B and (e) A? If the mass were doubled, would the change in the gravitational pot. energy of the system between points A and B increase, decrease, or remain the same? 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 14/8 SP 8-7 In the figure, a 2.0 kg package of tamale slides along a floor with speed v1 = 4.0 m/s. It then runs into and compresses a spring, until the package momentarily stops. Its path to the initially relaxed spring is frictionless, but as it compresses the spring, a kinetic frictional force from the floor, of magnitude 15 N, acts on it. The spring constant is 10,000 N/m. By what distance d is the spring compressed when the package stops? 5-Nov-06 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 15/8