Work done in lifting an object y = h A force F lifts the mass at constant speed through a height h. F The displacement is h a = 0 The applied force in the direction of the y = 0 displacement is: F = mg (no acceleration) mg The work done by the force F is: W = Fh = mgh But the kinetic energy has not changed – the gravity force mg has done an equal amount of negative work so that the net work done on the mass is zero 14 Work done in lifting an object y = h Alternative view: define a different form of energy – Gravitational potential energy, PE = mgy F a = 0 Define: Mechanical energy = kinetic energy + potential energy y = 0 Mechanical energy, E = mv2/2 + mgy mg Then: Work done by applied force, F, is (change in KE) + (change in PE) So W = Fh = ∆KE + ∆PE 15 Check, using forces and acceleration y = h v Net upward force on the mass is F – mg F Apply Newton’s second law to find the acceleration: a F – mg = ma, y = 0 vo so, a = (F – mg)/m mg One of famous four equations – 2 2 v = v0 + 2ah So, v2 = v2 + 2(F mg)h/m 0 − ( m/2) × mv2/2 mv2/2 = Fh mgh That is, DKE = Fh DPE − 0 − − Or, W = Fh = DKE + DPE 16 Or, W = Fh = DKE + DPE If there is no external force, W = 0 and 0 = DKE + DPE so that DKE = DPE − As the mass falls and loses potential energy, it gains an equal amount of kinetic energy. Potential energy is converted into kinetic. Energy is conserved overall. 17 6.26 m = 0.6 kg, yo = 6.1 m Ball is caught at y = 1.5 m mg a) Work done on ball by its weight? Weight force is in same direction as the displacement so, Work = mg × displacement = 0.6g × (6.1 - 1.5 m) = 27 J b) PE of ball relative to ground when released? PE = mgyo = 0.6g × (6.1 m) = 35.9 J c) PE of ball when caught? PE = mgy = 0.6g × (1.5 m) = 8.8 J 18 d) How is the change in the ball’s PE related to the work done by its weight? Change in PE = mg(y - yo) (final minus initial) Work done by weight = mg ×(displacement) = mg(yo - y) = – ∆PE 19 Conservation of Mechanical Energy In the absence of applied forces and friction: Work done by applied force = 0 So, 0 = (change in KE) + (change in PE) And KE + PE = E = constant 20 Example: No applied (i.e. external) forces E = KE + PE = constant KE = mv2/2 PE = mgy So E = mv2/2 + mgy = constant, until the ball hits the ground 21 Check: vx = v0 cosq = constant, in absence of air resistance v2 = v2 2g(y y ) = (v sinq)2 2g(y y ) when object is at height y y 0y − − 0 0 − − 0 v2 = v2 + v2 = (v cosq)2 + (v sinq)2 2g(y y ) x y 0 0 − − 0 2 2 2 2 v = v0 2g(y y0), as sin q + cos q = 1 ( m/2) − − × 2 2 So, mv0/2 + mgy0 = mv /2 + mgy and KE + PE = constant 22 6.34 Find the maximum height, H. Ignore air resistance. Conservation of mechanical energy: KE + PE = constant 2 At take-off, set y = 0: E = mv0/2 + 0 At highest point, y = H: E = mv2/2 + mgH 2 2 So, E = mv0/2 = mv /2 + mgH (v2 v2)/2 (142 132)/2 H = 0 − = − = 1.38 m g 9.8 23 6.38 v = 0 at highest point y = = yo Find the speed of the particle at A (vo). There is no friction. Conservation of mechanical energy: E = KE + PE = constant 2 2 At A: E = mv0/2 + mgy0 = mv0/2 + 3mg At highest point: E = KE + mgy = 0 + 4mg 24 2 2 At A: E = mv0/2 + mgy0 = mv0/2 + 3mg At highest point: E = KE + mgy = 0 + 4mg 2 So, E = mv0/2 + 3mg = 4mg 2 mv0/2 = mg v0 = 2g = 4.43 m/s ! What happens at B doesn’t matter, provided there is no loss of energy due to friction! 25 Conservative Forces Gravitational potential energy depends only on height The difference in PE, mg(ho - hf) is independent of path taken Gravity is a “conservative force” 26 Conservative Forces Alternative definitions of conservative forces: • The work done by a conservative force in moving an object is independent of the path taken. • A force is conservative when it does no net work in moving an object around a closed path, ending up where it started. In either case, the potential energy due to a conservative force depends only on position. Examples • Gravity • Elastic spring force • Electric force 27 Non-conservative Forces The work done by a non-conservative force depends on the path. Friction – the longer the path taken, the more the (negative) work done by the friction force. Defining a potential energy relies on a conservative force so that the work done in moving an object from A to B depends only on the positions of A and B. Examples of non-conservative forces • Static and kinetic friction forces • Air resistance • Tension, or any applied force • Normal force • Propulsion force in a rocket 28.