1. PHY 231 1 PHYSICS 231 Lecture 10: Too much work! Remco Zegers Walk-in hour: Thursday 11:30-13:30 Helproom

2. PHY 231 2 WORK Work: ‘Transfer of energy’ Quantitatively: The work W done by a constant force on an object is the product of the force along the direction of displacement and the magnitude of displacement. W=(Fcos  )  x Units: =Nm=Joule

3. PHY 231 3 An example xx n n FgFg FgFg T T The work done by the person on the suitcase: W=(Tcos45 o )  x The work done by F g on the suitcase: W=(F g cos270 o )  x=0 The work done by n on the suitcase: W=(F g cos90 o )  x=0 The work done by friction on the suitcase: W=(f k cos180 o )  x=- u k n  x The work done by the suitcase on the person: W=(Tcos225 o )  x  =45 o fkfk opposite

4. PHY 231 4 Non-constant force W=(Fcos  )  x: what if Fcos  not constant while covering  x? Example: what if  changes while dragging the suitcase? Fcos  Area=  A= (Fcos  )  x The work done is the same as the area under the graph of Fcos  versus x W=  (  A)

5. PHY 231 5 Power: The rate of energy transfer Work (the amount of energy transfer) is independent of time. W=(Fcos  )  x … no time in here! To measure the how fast we transfer the energy we define: Power(P)=W/  t (J/s=Watt) (think about horsepower etc). P =(Fcos  )  x/  t=(Fcos  )v average

6. PHY 231 6 Example A toy-rocket of 5.0 kg, after the initial acceleration stage, travels 100 m in 2 seconds. What is the work done by the engine? What is the power of the engine? h=100m

7. PHY 231 7 Potential Energy Potential energy (PE): energy associated with the position of an object within some system. Gravitational potential energy: Consider the work done by the gravity in case of the rocket: W gravity =F g cos(180 o )  h=-mg  h=-(mgh f -mgh i )=mgh i -mgh f =PE i -PE f The ‘system’ is the gravitational field of the earth. PE=mgh Since we are usually interested in the change in gravitational potential energy, we can choose the ground level (h=0) in a convenient way.

8. PHY 231 8 Another rocket h=100m A toy rocket (5kg) is launched from rest and reaches a height of 100 m within 2 seconds. What is the work done by the engine during acceleration?

9. PHY 231 9 Kinetic energy Consider object that changes speed only a) W=F  x=(ma)  x … used Newton’s second law b) v=v 0 +at so t=(v-v 0 )/a c) x=x 0 +v 0 t+0.5at 2 so x-x 0 =  x=v 0 t+0.5at 2 Combine b) & c) d) a  x=(v 2 -v 0 2 )/2 Combine a) & d) W=½m(v 2 -v 0 2 ) Kinetic energy: KE=½mv 2 When work is done on an object and the only change is its speed: The work done is equal to the change in KE: W=KE final -KE initial  x=100m V=0  t=2s

10. PHY 231 10 Conservation of mechanical energy Mechanical energy = potential energy + kinetic energy In a closed system, mechanical energy is conserved * ME=mgh+½mv 2 =constant * There is an additional condition, see slides 12,13,14

11. PHY 231 11 Example of closed system A snowball is launched horizontally from the top of a building at v=12.7 m/s. The building is 35m high. The mass is 0.2 kg. Is mechanical energy conserved? h=35m d=34m V 0 =12.7 m/s

12. PHY 231 12 Conservative forces A force is conservative if the work done by the force when Moving an object from A to B does not depend on the path taken from A to B. Example: gravitational force h=10m Using the stairs: W g =mgh f -mgh i =mg(h f -h i ) Using the elevator: W g =mgh f -mgh i =mg(h f -h i ) The path taken (longer or shorter) does not matter: only the displacement does!

13. PHY 231 13 Non conservative forces A force is non-conservative if the work done by the force when moving an object from A to B depends on the path taken from A to B. Example: Friction You have to perform more work Against friction if you take the long path, compared to the short path. The friction force changes kinetic energy into heat. Heat, chemical energy (e.g battery or fuel in an engine) Are sources or sinks of internal energy.

14. PHY 231 14 Conservation of mechanical energy only holds if the system is closed AND all forces are conservative ME i -ME f =(PE+KE) i -(PE+KE) f =0 if all forces are conservative Example: throwing a snowball from a building neglecting air resistance ME i -ME f =(PE+KE) i -(PE+KE) f =W nc if some forces are nonconservative. W nc =work done by non-conservative forces. Example: throwing a snowball from a build taking into account air resistance