There are two types of forces: conservative (gravity, spring force) nonconservative (friction)

Transcription

1 Chapter 8: Conservation o Energy There are two types o orces: conservative (gravity, spring orce) nonconservative (riction)

2 Conservative Forces Conservative Force the work done by the orce on an object moving rom one point to another depends only on the initial and inal positions and is independent o the particular path taken. or Conservative ve Force the net work done by a orce on an object moving around a closed path is zero. Consider the work done by ygravity: W W G G F G dl 1 y y 1 mg dy 1 mg cos θ dl mg ( y y ) 1

3 Conservative Forces A orce is conservative i the work it does on an object moving between two points is independent o the path taken. work done depends only on r i and r I an object moves in a closed path (r i r ) then total work done by the conservative orce is zero.

4 Consider riction Nonconservative Forces work done by the orce depends on the path i non-conservative orces dissipate energy

5 Potential Energy Potential lenergy in a system is associated with the position or coniguration o objects in the system. For example: When you lit a ball a distance y, the gravitational orce does negative work on the ball. (The new position o the ball relative to the earth is changed by y. The energy that was stored in this new coniguration o the earth-ball system is called gravitational potential energy. It turns out that, W c ΔU (U U i ) W c work done by a conservative orce U potential energy The zero o potential energy is arbitrary. Only the change in potential energy is relevant.

6 Gravitational Potential Energy The gravitational orce is conservative! g ( 0) W mg y mgy ( ) W U U U + 0 g y 0 y U y mgy We call this potential energy because i this mass is released, and allowed to all, it can do work. Potential energy is thus stored in the system and available to do work i released.

7 Example When a 4-kg object is moved rom the ground to a shel 1 m high, what is the change in its potential energy? What is the change in potential energy i the same object is moved rom the 1 m shel to a shel m high? Δ U mgy mgy ( ) 1 i J Δ U mgy mgy ( ) i J

8 Spring Potential Energy The potential energy stored in a stretched or compressed spring is, U ½ kx x displacement rom equilibrium position Notice the sign o x, i.e., + or doesn t matter or the potential energy.

9 Example An 8 kg mountain climber is in the inal stage o the ascent o 4301 m Pikes Peak. What is the change in gravitational potential energy as the climber gains the last 100 m o altitude?

10 Let U i 0 at sea level, l ΔU mgy mgy i (8kg)(9.8m/s )(4301m) (8kg)(9.8m/s )(401m) ΔU 80,400J LtU Let U i 0 at 4301m, ΔU mgy mgy i (8 kg )(9.8m/s )(0 m ) (8 kg )(9.8m/s )( 100m ) ΔU 80,400J

11 Potential Energy Summarized A potential energy is always associated with a conservative orce, and the dierence in potential energy between two points is deined as the negative o the work done by that orce. The choice o where U0 is arbitrary and can be chosen wherever it is most convenient. Since a orce is always exerted on one body by Since a orce is always exerted on one body by another body potential energy is associated with the interaction o two or more bodies.

12 Conservation o Energy Energy is neither created nor destroyed d The energy o an isolated system o objects remains constant.

13 Mechanical Energy Mechanical energy E is the sum o the potential and kinetic energies o an object. E U+K The total mechanical energy in any isolated system o objects remains constant i the objects interact only through conservative orces: E constant E E i U + K U i + K i

14 Example A 5.00-kg rock is dropped and allowed to all reely. Find the speed o the ball ater it has dropped m.

15 Deine the release point as U0. Write down the conservation o energy theorem. KE + U KE + U 0 + i 0 i 1 (5kg) v + (5kg)(9.8m / s )( m) v 6.3m / s

16 Pendulum A 0.-kg pendulum bob is swinging back and orth. I the speed o the bob at its lowest point is 0.65 m/s, how high does the bob go above its minimum height? h

17 Use conservation o energy, KE + U KE + U i i i 1 mv mgh ( 0.65m/s) v h i. g (9.8m/s ) cm

18 Energy Conservation with Dissipative Forces: Solving Problems Remember the work-energy theorem: W net ΔKE Also, remember that the work done by conservative orces is: W c ΔU But, W W + W net c nc So, Δ KE Δ U + W Wnc W nc ΔKE + ΔU

19 Example A kg ball is dropped rom a height o 5m. The work done on the ball by riction is W -00 J. How ast is the ball moving when it strikes the ground? 1 1 Δ KE mv mv Δ U + W ( mgh mgh ) + W 1 (kg) v i nc 0 (0 ( kg)(9.8m / s v 17 m / s )(5m)) + ( 00J ) i Without air riction v m/s

20 Power Power is deined as the amount o work done per unit time. The average power over a time interval t is, P W t SI Units: watt joule/second Power can also be written in terms o velocity and displacement: dw d dl P ( F l ) F dt dt dt P F v What s the condition on F here?

21 Examples 4. A 66.5 kg hiker starts at an elevation o 1500 m and climbs to the top o a 660 m peak. (a) What is the hiker s change in potential energy? (b) What is the minimum i work required by the hiker? (c) Can the actual work done be greater than this? 8. Air resistance can be represented by a orce proportional to the velocity V o an object: F-kv. Is this orce conservative? Explain. 13. In the high jump, the kinetic energy o an athlete is transormed dit into gravitational itti potential tilenergy without tthe aid o a pole. With what minimum speed must the athlete leave the ground in order to lit his center o mass.10 m and cross the bar with a speed o 0.70 m/s?

22 Examples Cont. Calculate the power required o a 1400 kg car under the ollowing circumstances: (a) () the car climbs a 10 o hill at a steady 80 km/h; and (b) the car accelerates along a level road rom 90 to 110 km/h in 6 seconds to pass another car. Assume that the car has a retarding orce o F R 700 N throughout (this is due to air resistance and riction inside the wheel bearings).

23 Example A 1.9-kg block slides down a rictionless ramp, as shown in the Figure. The top o the ramp is 1.5 m above the ground; the bottom o the ramp is h m above the ground. The block kleaves the ramp moving horizontally, and lands a horizontal distance d away. Find the distance d.

24 Use conservation o mechanical to ind the velocity at the bottom o the ramp. Ki + Ui K + U 1 + mgh mv + v gh 0 0 Now we have a projectile problem. 1 x ght y 0.5 gt gt to hit t to hit 9.81 t 0.6 s to hit ( ) ( ) d gh tto hit m

25 Additional Examples 6. I U 3x +xy+4y z, what is the orce, F? 15. A 60 kg bungee jumper jumps rom a bridge. She is ties to a bungee cord that is 1 m long when unstretched, and she alls a total o 31m. (a) Calculate the spring constant k o the bungee cord assuming Hooke s law applies. (b) Calculate the maximum acceleration elt by the jumper. 1. A pendulum.00m long is released (rom rest) at an angle o 30. Determine the speed o the g bob (a) at the lowest point (θ0 ); (b) at θ15.0 (c) θ-15.0 (d) Determine the tension in the cord at each o these three points. (e) I the speed o the bob is given an initial speed o vo1.0 m/s when release at 30.0, recalculate the speeds or parts (a), (b) and (c). 31. A skier traveling 11 m/s reaches the oot o a steady upward 17 incline and glides 1m up along the slope beore coming to a rest. What is the average coeicent o kinetic riction? 54. How long will it take a 1750 motor to lit a 85 kg piano to a sixth story window 16 m above? 58. What minimum horsepower must a motor have to be able to drag a 300 kg box along a level loor at a speed o 1. m/s i the coeicient o kinetic riction is 0.45?