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Summary Lecture 7 7.1-7.6 Work and Kinetic energy 8.2 Potential energy 8.3 Conservative Forces and Potential energy 8.5 Conservation of Mech. Energy 8.6 Potential-energy curves 8.8 Conservation of Energy Systems of Particles 9.2 Thursdays 12 – 2 pm PPP “ExtEnsion” lecture. Centre of mass Room 211 podium Problems: Chap. 8 5, 8, 22, 29, 36, 71, 51 level Turn up any time Chap. 9 1, 6, 82 Outline Lecture 7 Work and Kinetic energy Work done by a net force results in kinetic energy Some examples: gravity, spring, friction Potential energy Work done by some (conservative) forces can be retrieved. This leads to the principle that energy is conserved Conservation of Energy Potential-energy curves The dependence of the conservative force on position is related to the position dependence of the PE F(x) = -d(U)/dx Kinetic Energy Work-Kinetic Energy Theorem Change in KE work done by all forces DK Dw Work-Kinetic Energy Theorem Vector sum of all forces acting on the body SF w xx F .dx f i xx ma.dx f i x x i f dv m .dx dt x x i f dx m .dv dt xi xf m v .dv m[1 / 2v ] 2 vf vi vf vi = 1/2mvf2 – 1/2mvi2 = Kf - Ki = DK Work done by net force = change in KE x Gravitation and work Work done by me (take down as +ve) h F mg Lift mass m with constant velocity = F.(-h) = -mg(-h) = mgh Work done by gravity = mg.(-h) = -mgh ________ Total work by ALL forces (DW) = 0 =DK Work done by ALL forces = change in KE DW = DK What happens if I let go? Compressing a spring Compress a spring by an amount x F -kx x Work done by me Fdx = kxdx = 1/2kx2 Work done by spring -kxdx =-1/2kx2 Total work done (DW) = 0 =DK What happens if I let go? Moving a block against friction at constant velocity f F d Work done by me = F.d Work done by friction = -f.d = -F.d Total work done What happens if I let go? = 0 NOTHING!! Gravity and spring forces are Conservative Friction is NOT!! Conservative Forces A force is conservative if the work it does on a particle that moves through a round trip is zero: otherwise the force is non-conservative A force is conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points: otherwise the force is non-conservative. Conservative Forces A force is conservative if the work it does on a particle that moves through a round trip is zero; otherwise the force is non-conservative Consider throwing a mass up a height h -g h work done by gravity for round trip: On way up: work done by gravity = -mgh On way down: work done by gravity = mgh Total work done Sometimes written as F.ds 0 = 0 Conservative Forces A force is conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points: otherwise the force is non-conservative. Work done by gravity Each step height=Dh w = -mgDh1+ -mgDh2+-mgDh3+… = -mg(Dh1+Dh2+Dh3 +……) -g h = -mgh Same as direct path (-mgh) Potential Energy The change in potential energy is equal to minus the DU = -Dwforce ON the body. work done BY the conservative Work done by gravity h = mg.(-h) = -mgh Therefore change in PE is mg Lift mass m with constant velocity DU = -Dw DUgrav = +mgh Potential Energy The change in potential energy is equal to minus the work done BY the conservative force ON the body. Compress a spring by an amount x F -kx x Work done by spring is Dw = -kx dx = - ½ kx2 Therefore the change in PE is DU = - Dw DUspring = + ½ kx2 Potential Energy The change in potential energy is equal to minus the work done BY the conservative force ON the body. DU = -Dw but recall that Dw = DK so that DU = -DK or DU + DK = 0 Any decrease increase in PE results from a ndecrease increase in KE DU + DK = 0 In a system of conservative forces, any change in Potential energy is compensated for by an inverse change in Kinetic energy U+K=E In a system of conservative forces, the mechanical energy remains constant Potential-energy diagrams Dw = - DU = F. Dx thus DU F Dx In the limit dU F dx The force is the negative gradient of the PE curve If we know how the PE varies with position, we can find the conservative force as a function of position PE of a spring dU F dx Energy U= ½ kx2 here U = ½ kx2 dU so F dx d 1 2 ( 2 kx ) dx F kx x Energy Potential energy U= ½ kx2 Total mech. energy At any position x PE + KE = E KE U= ½ kA2 E= U+K=E K=E-U = ½ kA2 – ½ kx2 = ½ k(A2 -x2) PE x’ x x=A Roller Coaster K Fnet=-dU/dt Et U Fnet = mg – R R = mg - Fnet K Et U R Fnet=-dU/dx mg Fnet = mg – R R = mg - Fnet Conservation of Energy We said: when conservative forces act on a body DU + DK = 0 U + K = E (const) This would mean that a pendulum would swing for ever. In the real world this does not happen. Conservation of Energy When non-conservative forces are involved, energy can appear in forms other than PE and KE (e.g. heat from friction) Energy converted to int other forms DU + DK + DU =0 Ki + Ui = Kf + Uf + Uint Energy may be transformed from one kind to another in an isolated system, but it cannot be created or destroyed. The total energy of the system always remains constant. Stone thrown into air, with air resistance. How high does it go? Ei f mg = Ef + Eloss Ki + Ui = Kf + Uf + Eloss ½mvo2 + 0 = 0 + mgh + fh h ½mvo2 = h(mg + f) v0 upward 2 0 mv h 2(mg f) Stone thrown into air, with air resistance. What is the final velocity ? E’i = E’f + E’loss f K’i + U’i = K’f + U’f + E’loss 0+ mg mgh = mv 02 2(mg f) ½mvf2 = ½mvf2 +0 = ½mvf2 + f mv 02 mg 2(mg f) - f mv 02 + fh 2(mg f) mv 02 2(mg f) mg h mv 02 2(mg f) v 02 mg f v (mg f) 2 f mg f v v mg f 2 f 2 0 downward Centre of Mass (1D) M = m 1 + m2 M m1 0 x1 m2 x2 xcm M xcm = m1 x1 + m2 x2 xcm m1x1 m 2x 2 M In general xcm 1 m i xi M