Chapter 8: Conservation o Energy There are two types o orces: conservative (gravity, spring orce) nonconservative (riction)
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
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.
Consider riction Nonconservative Forces work done by the orce depends on the path i non-conservative orces dissipate energy
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.
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.
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 4 9.81 1 0 39.4 J Δ U mgy mgy ( ) i 4 9.81 1 39.4 J
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.
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?
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
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.
Conservation o Energy Energy is neither created nor destroyed d The energy o an isolated system o objects remains constant.
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
Example A 5.00-kg rock is dropped and allowed to all reely. Find the speed o the ball ater it has dropped m.
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
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
Use conservation o energy, KE + U KE + U i i i 1 mv + 0 0 + mgh ( 0.65m/s) v h i. g (9.8m/s ) cm
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
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
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?
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?
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).
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 05 0.5 m above the ground. The block kleaves the ramp moving horizontally, and lands a horizontal distance d away. Find the distance d.
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 1 0.5 0 0.5 gt to hit t to hit 9.81 t 0.6 s to hit ( ) ( ) d gh tto hit 9.81 1.5 0.5 0.6 1.1 1 m
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 70.0 0 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?