Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-061/8 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 Chapter 8: Potential Energy and Conservative Forces

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-062/8 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, …) 8. 2 Work and Potential Energy

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-063/8 We know that the work done result in a change in kinetic energy (W=  K). 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 F c ) does some work (W c ) by changing the system configuration, the force derives the energy from the stored potential energy associated with that force:

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-064/8 Conservative and Nonconservative forces We can define a potential energy for this force by the equation  U=-W only if W 12 = - W 21. A force for which W 12 = - W 21 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.

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-065/8 The gravitational force and the spring force are examples of conservative forces. The frictional force and fluid drag force are examples of nonconservative forces.

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-066/8 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

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-067/8 In general we can write: DO CP 8-2 Gravitational potential energy,  U g 8.4 Determining Potential Energy Values

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-068/8 x i x f Elastic potential energy stored in a spring: The spring is stretched or compresses from its equilibrium position by x Elastic potential energy,  U s

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-069/8 SP 8-2 (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  U in the potential energy of the sloth-Earth system due to the fall?

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-0610/8 If we deal only with conservative forces and If we deal with an isolated system (no energy added or removed): The total mechanical energy of a system remains constant!!!! The final and initial energy of a system remain the same: E i = E f Thus: 8.5 Conservation of Mechanical Energy

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-0611/8 Figure 8-7

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-0612/8 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: And also add nonconservative forces (friction involved) in the system: (increase in thermal energy by sliding) (work done on the system, friction involved)

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-0613/8 8-8 Conservation of Energy The total energy of a system can change only by amounts of energy W ex that are transferred to or from the system.  E int 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. Power as the rate at which energy is transferred from one form to another 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).

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-0614/8 A frictionless roller coaster of mass m 825 kg tops the first hill with v 0 = 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? P.5

Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS 5-Nov-0615/8 SP 8-7 In the figure, a 2.0 kg package of tamale slides along a floor with speed v 1 = 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?