Conservation of Energy 1 of 8 Conservation of Energy The important conclusions of this chapter are: • If a system is isolated and there is no friction (no non-conservative forces), then KE + PE = constant (Our text uses the notation K +U = constant) • If there is friction, then KE + PE + Etherm = constant. (Etherm = thermal energy) • Two examples of PE (potential energy) PEgrav = mgh PEelastic = (1/2)kx2 Questions you should have: Why is KE+PE = constant, when system isolated and no friction? What is the definition of potential energy, PE, and why PEgrav = mgh, PEelastic = (1/2)kx2 ? It is not enough to know formulas. You should know where the formulas come from. Potential Energy So, how do we define potential energy, PE, and get PEgrav = mgh ? If a force involves no dissipation (no friction), then it iusually a special type of force called a conservative force. The defining property of a conservative force is that the work done by a this force depends only the initial and final positions, not on the path taken. We showed in a previous concept test, that gravity is a conservative force. The force of friction is not a conservative force, because the work done depends on the path taken: the longer the path the more work is done by friction. s i f 2/24/2009, PHYS1110 Notes Dubson ©University of Colorado at BoulderConservation of Energy 2 of 8 Conservative forces include: • gravity (F = mg) • the spring force, or elastic force (F = −kx) • the normal force The normal force is something of a special case. The work done by the normal force is always zero, so the normal force is "trivially" path-independent: the work is zero, regardless of path, and regardless of initial and final positions. Associated with every conservative force is a kind of energy called potential energy (PE or U). PE is a kind of stored energy. If a configuration of objects has PE, then there is the potential to change that PE into other kinds of energy (KE, thermal, light, etc ). The definition of the PE associated with a conservative force involves the work done by that force. Let’s first review. Recall: If I lift a mass m, a distance h, at constant velocity (v = constant), with an external force Fext , such as my hand, then the work done by gravity is the negative of the work done by the external force. f Fext = mg So Wext = +mgh and Wgrav = −mgh . This is true for the special case v = constant, but it turns out that it is always true that Wext = −Wgrav , regardless of the motion, so long as there is no change in the KE. So, Wext = −Wgrav , if the mass starts and finishes at rest: vi = vf = 0. With this example in mind, we are ready to define PE. If a force F (such as gravity) is a conservative force, then we define the PE associated with that force by FFextPE W W∆≡−=+ In words: the change in potential energy is the negative of the work done by the conservative force and it is the positive of the work done by an external force opposing the conservative force. h i Fgrav = mg same magnitudes, opposite directions 2/24/2009, PHYS1110 Notes Dubson ©University of Colorado at BoulderConservation of Energy 3 of 8 Only changes in PE are physically meaningful. We are free to set the zero of potential wherever we want. grav extPE W mgh∆ =+ =. In this formula, the height h is the height above (h+) or below (h−) the h=0 level. So h is really . Nfi f0hhh h∆ = − ==h=+ So I should really write the formula as . gravPE mg h∆ = ∆ If I choose to set PEi = 0 at hi = 0, then the formula becomes or simply, gravPE mg h∆ = ∆NNii00PE PE mg(h h )− = −gravPE mgh= In the previous chapter, we showed that the work done by an external force to stretch or compress a spring by an amount x is Wext = +(1/2)kx2. We therefore have that the elastic potential energy contained in a spring is ∆ elas extPE W 21elas2PE k x∆ = In writing this formula, we have set PEelas = 0 at x = 0 (the unstretched position). (The normal force never does work, so ∆. We can set the PE associated with the normal force equal to zero and forget about it.) normal normalPE W 0= − = Where is potential energy located? I lift a book of mass m a height h and say that the book has PEgrav = mgh. But it is not correct to say that the PE “in the book”. The gravitational PE is associated with the system of (book + earth + gravitational attraction between book and earth). The PE is not "in the book" or "in the earth"; it is in the book-earth system which includes the “gravitational field” surrounding the book and the earth. For the case of elastic potential energy, the PEelas actually is inside the spring. It is located in the increased electrostatic potential energy in the chemical bonds joining the atoms of the spring. Conservation of mechanical energy. Definition: mechanical energy Emech = KE + PE. We are now in a position to show that Emech = KE + PE = constant (if no friction and system isolated). 2/24/2009, PHYS1110 Notes Dubson ©University of Colorado at BoulderConservation of Energy 4 of 8 Recall the Work-KE Theorem: Wnet = ∆KE. Now, if there is no friction, the net force involves conservative forces only, so Wnet = Wc (c for conservative force). But we just defined , so we have or cPE W∆≡−net cWW KE P==∆ = −∆ EKE PE 0 KE PE constant∆ + ∆ = ⇔ += (if no friction) Example of Conservation of Energy (no friction). A pendulum consists of a mass m attached to a massless string of length L. The pendulum is released from rest a height h above its lowest point. What is the speed of the pendulum mass when it is at height h/2 from the lowest point? Assume no dissipation (no friction). In all Conservation of Energy problems, begin by writing (initial energy) = (final energy) : LE i = E f ⇒ KEi + PEi = KEf + PEf ⇒ 0 + mgh = (1/2) mv2 + mg(h/2) (cancel m's and multiply through by 2) ⇒ 2gh = v2 + gh ⇒ v2 = gh ⇒ vgh= Notice: Using Conservation of Energy, we didn't need to know anything about the details of the forces involved and we didn't need to use Fnet = ma. The Conservation of Energy strategy allows us to relate conditions at the beginning to conditions at the end; we don't need to know anything about the details of what goes on in between. Suppose there are two conservative forces acting on a system, and no non-conservative forces. Then we have