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conservation_principles

conservation_principles - CONSERVATION PRINCIPLES CPM...

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Unformatted text preview: CONSERVATION PRINCIPLES CPM: CONSERVATION OF MASS IN A CLOSED (NO EXCHANGE OF MATTER WITH SURROUNDIN GS) SYSTEM THE TOTAL MASS IS CONSTANT. ' CPE: CONSERVATION OF ENERGY IN AN ISOLATED SYSTEM TOTAL, .3‘ ENERGY IS CONSTANT. IN OUR PRESENT DISCUSSION WE TALK OF MECHANICAL ENERGY. CPP: CONSERVATION OF LINEAR MOMENTUM . IF THERE IS NO EXTERNAL FORCE PRESENT, THE TOTAL (VECTOR) LINEAR MOMENTUM OF A SYSTEM IS CONSTANT. CPL: CONSERVATION OF ANGULAR MOMENTUM IF THERE IS NO EXTERNAL TORQUE THE TOTAL (VECTOR) ANGULAR MOMENTUM IS CONSTANT. CPE The ingredients required to state the conservation principle for mechanical energy are: MECHANICAL WORK If the point of application of a constant force E is displaced through an amount gs: the amount of work done is _ W = E - AS = FAS cos(_E, AIS) =15;ch +FyAy+FzAz . so AW is the “DOT” product of the force vector and the displacement vector. Notice, that we are multiplying the component of 1]: along AS: by AS to get the Work done. No work is DONE if E L $51. Also, note that _A_,S measure the total displacement of E . For example, AB is half a circle of radius R. If you apply a force E which is tangent to EB at every point, total work done is AW = F 7zR F Wu? If Fx , varies with x, work done is are under Fx vs. x curve. ‘ F M A. Ex: Spring Force 1"; = “ kKQL. ‘ Work done by spring in going from x to zero is AW=lkx2 2 KINETIC ENERGY MECHANICAL WORK STORED IN GIVING A FINITE SPEED TO A MASS— WORK STORED IN MOTION Object at rest as x0 Apply force F: Fx, keep force on until object reaches x C31 x0, AW=F(x-—xo) =Md(x-x0) But we know that v2 = v02 + 2a(x—- x0) v0 :0; v=2a(§c¥xo) sci AW=1Mv 2 After F turned off, M moved on with speed v We have stored kinetic energy K = lez 2 in its motion. POTENTIAL ENERGY (See attached note) . MECHANICAL WORK STORED 'IN A SYSTEM WHEN IT IS ASSEMBLED IN THE PRESENCE OF A PREVAILIN G CONSERVATIVE FORCE E . Change in potential energy AP: —F . AS aa- (NEVER FORWT THE “MINUS” SIGN! WHY?) We have two conservative forces available i) Wg = —ng2, so taking Pg = 0 at Earth’s surface we write Pg ()0 = ng as the potential energy of the Mass-Earth system. ii) FSP = 4066 so 1 Rrpcx) :51“: Now we have all three W, (It? and P we can write CPE. ISOLATED SYSTEM K1. + Pf = K, + 13 1': initial state f = final state EXTERNAL WORK INCLUDED Kf+Pf:K,.+P,+WNCF cm NCF refers to Non-Conservative force. (friction, force applied by your hzeadetc.) Note: if NCF is fk, WW IS ALWAYS NEGATIVE!!! Potential Energy (P) presents a greater conceptual challenge. P is the mechanical work stored in a system when it is prepared (or put together) in the presence of a prevailing conservative force. ' ' Suppose we have a region of space in which there is a prevailing force (weight near Earth’s surface comes to mind). That is, at every point in this region an object will experience a force. Let the object be at point B [First, noticethat you can’t let the object go as E will immediately cause 95 and object will move]. To define P at B we have to calculate how much work was needed to put the object at B in the presence of E . Le us pick some point A, where we can claim that P is known, and calculate the work needed to go from A to B. As soon as we try to do that we realize that the only way we can get a meaningful answer is if the work required to go from A to B is independent of the path taken. So our prevailing force has to be special; Such a force is called a CONSERVATIVE FORCE — WORK DONE DEPENDS ONLY ON END-POINTS AND NOT ON THE PATH TAKEN. If that is true we. have a unique answer AW, = AW2 = AW3 = AWAB and we can use this fact to calculate the change in P in going from A to B AP AB = ”E ° ASAB NOTE THE ~SIGN: It comes about because as stated above we cannot let the object go. In fact, _ the displacement from A to B must be carried out in such a way that the object cannot change its speed (if any). That is, we need to apply a force —— E to balance the ambient E at every point. The net force will become close to zero at all points. APAB is work done by~ E . So when E is conservative APAB is unique. In the final step we can choose A such that PA = 0. ...
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