Chapter 8 Conservation of energy I Work done on a system by an external force II Conservation of mechanical energy II External work and thermal energy III External forces and internal energy changes IV Power I Work done on a system by an external force Work is energy transfer to or from a system by means of an external force acting on that system When more than one force acts on a system their net work is the energy transferred to or from the system No Friction Remember W Emec K U Ext force Emec K U 0 only when System isolated No ext forces act on a system All internal forces are conservative Friction F f k ma v 2 v02 2ad a 0 5 v 2 v02 d m 2 2 1 1 1 v v0 Fd f k d m v 2 v02 Fd mv 2 mv02 f k d 2 2 2 2d W Fd K f k d F fk General W Fd Emec f k d Example Block sliding up a ramp Thermal energy E f d th k Friction due to cold welding between two surfaces As the block slides over the floor the sliding causes tearing and reforming of the welds between the block and the floor which makes the block floor warmer Work done on a system by an external force friction involved W Fd Emec Eth II Conservation of energy Total energy of a system E mechanical E thermal E internal The total energy of a system can only change by amounts of energy transferred from or to the system W E Emec Eth Eint Experimental law The total energy of an isolated system cannot change There cannot be energy transfers to or from it Isolated system Emec Eth Eint 0 In an isolated system we can relate the total energy at one instant to the total energy at another instant without considering the energies at intermediate states Example Trolley pole jumper 1 Run Internal energy muscles gets transferred into kinetic energy 2 Jump Ascent Kinetic energy transferred to potential elastic energy trolley pole deformation and to gravitational potential energy 3 Descent Gravitational potential energy gets transferred into kinetic energy III External forces and internal energy changes Example skater pushes herself away from a railing There is a force F on her from the railing that increases her kinetic energy i One part of an object skater s arm does not move like the rest of body ii Internal energy transfer from one part of the system to another via the external force F Biochemical energy from muscles transferred to kinetic energy of the body WF ext K F cos d Non isolated system K U WF ext Fd cos Emec Fd cos Change in system s mechanical energy by an external force Proof v 2 v02 2a x d 0 5M 1 1 Mv 2 Mv02 Ma x d 2 2 K F cos d IV Power Average power Pavg Instantaneous power E t P dE dt 61 In the figure below a block slides along a path that is without friction until the block reaches the section of length L 0 75m which begins at height h 2m In that section the coefficient of kinetic friction is 0 4 The block passes through point A with a speed of 8m s Does it reach point B where the section of friction ends If so what is the speed there and if not what greatest height above point A does it reach N mg cos 30 8 5m f k k N 0 4 8 5m 3 4m N A C Only conservative forces Emec 0 f C mg K A U A KC U C 1 2 1 2 mv A mvc mghc vc 5m s 2 2 The kinetic energy in C turns into thermal and potential energy Block stops K c 0 5mvc2 12 4m K c mgy f k d 12 4m mg d sin 30 3 4md d 1 49 meters d L 0 75m Block reaches B Isolated system E 0 Emec U Eth K C U C K B U B f k L 12 4m 0 5mvB2 mg y B yc k mgL cos 30 0 5mvB2 mgL sin 30 k mgL cos 30 12 4m 0 5mvB2 3 67m 2 5m vB 3 5m s 129 A massless rigid rod of length L has a ball of mass m attached to one end The other end is pivoted in such a way that the ball will move in a vertical circle First assume that there is no friction at the pivot The system is launched downward from the horizontal position A with initial speed v0 The ball just barely reaches point D and then stops a Derive an expression for v0 in terms of L m and g b What is the tension in the rod when the ball passes through B c A little girl is placed on the pivot to increase the friction there Then the ball just barely reaches C when launched from A with the same speed as before What is the decrease in mechanical energy during this motion d What is the decrease in mechanical energy by the time the ball finally comes to rest at B after several oscillations a Emec 0 K f U f K i U i b Fcent mac T mg K D 0 U A 0 vB2 1 m T mg T m vB2 g L L U A K A UB KB mgL 1 2 mv0 v0 2 gL 2 c vc 0 W E Emec Eth Eth f k d 1 2 1 mv0 mgL mvB2 2 2 1 1 2 gL gL vB2 vB 2 gL 2 2 A L C x v0 T 5mg The difference in heights or in gravitational potential energies between the positions C reached by the ball when there is friction and D during the frictionless movement Is going to be the loss of mechanical energy which goes into thermal energy c Eth mgL d The difference in height between B and D is 2L The total loss of mechanical energy which all goes into thermal energy is Emec 2mgL D y T B Fc mg 101 A 3kg sloth hangs 3m above the ground a What is the gravitational potential energy of the sloth Earth system if we take the reference point y 0 to be at the ground If the sloth drops to the ground and air drag on it is assumed to be negligible what are b the kinetic energy and c the speed of the sloth just before it reaches the ground a Emec 0 K f U f K i U i b K f 94 1J U f ground 0 K i 0 c K f 2K f 1 2 7 67m s mv f v f 2 m U i mgh 3 2kg 9 8m s 2 3m 94 1J 130 A metal tool is sharpen by being held against the rim of a wheel on a grinding machine by a force of 180N …