Lecture 9. > Potential Energy. > Conservation of Energy. > Power. (Source: Serway; Giancoli) Villacorta--DLSUM-BIOPHY1-L Term01

Transcription

1 Lecture 9 > Potential Energy > Conservation of Energy > Power (Source: Serway; Giancoli) 1

2 Conservative & Nonconservative Forces > The various ways work and energy appear in some processes lead to two types of forces. + Forces that cause work on a body and still get back the energy lost + Forces that cause work on a body and dissipate energy away > Conservative forces allow objects to lose energy through work and gain it back again when the process is reversed. + Climbing up a diving board, jumping back down + Or sliding down from the same height: same energy for different paths > Nonconservative forces dissipate energy into the environment and cannot be gained back when the process is reversed. + Sliding an object across a table surface: direct path vs indirect path between the same points + The work done depends on the path between the points > Conservative forces are associated with an energy due to the state or position of the body: this is the potential energy. 2

3 Gravitational Potential Energy > When the Earth acts on a body above the surface, gravitational force is present between the two objects. + The body falls to the Earth due to this pull, resulting in work. + Gravitational work = gravitational force x path taken y i F g W g = F g d W g =m g ( y i y f ) W g =m g y i m g y f W g = (m g y f m g y i ) d = y i y f Let PE g =m g y y f W g = (PE gf PE gi ) W g = PE g F g Gravitational work = negative change in potential energy 3

4 Gravitational Potential Energy contd > Consider taking a different path between the same levels: W g '=(F g cosθ) d ' W g '=m g d ' cosθ y i But(d ' cosθ)= y i y f, then F g θ d' W g '=m g ( y i y f ) W g '= PE g W g '=W g d' cos θ y f > Gravitational work is the same for different paths between the same levels. F g > Gravitational work is pathindependent. 4

5 Gravitational Potential Energy contd > The gravitational work is related to the change in gravitational potential energy between different heights and not to the actual energy at a specific height. > The heights are commonly measured with respect to the surface of the Earth, where y = 0 and PE g = 0. + This choice is not always required and can be changed: any level can be chosen as reference for zero gravitational potential energy. + For problems involving gravitational potential energy, choose an appropriate reference at the start and stay with it during the analysis. 5

6 Elastic Potential Energy > Consider a spring initially not compressed nor stretched: Hooke's Law x 0 spring constant F s = k x = kx > A stretched spring is displaced by x and exerts a restoring force F s. + F s is opposite x: F s tries to bring the spring back to equilibrium + The larger x is, the greater F s. + Unlike F g, F s is not constant. > Computing for the work done by the spring requires the average force exerted over the displacement: W s = F x x = x x 0 = 0 x f = x F= F sf +F si 2 F= 1 2 k x = k x+0 2 6

7 Elastic Potential Energy contd > Thus, W s =( 1 2 k x) x > Consider the work done by a spring stretched from x i to x f : F= k ( x f + x i 2 ) W s = 1 2 k ( x)2 W s = 1 2 k x2 > The elastic work is negative since F s is always antiparallel to x. + Bodies attached to springs are more difficult to move from the equilibrium position. + F s keeps these attached bodies close to x 0. x 0 = 0 W s = F d W s =[ 1 2 k ( x f + x i )] ( x f x i ) x i d W s = 1 2 k ( x 2 f x 2 i ) x f 7

8 Elastic Potential Energy contd F= k ( x f + x i 2 ) Let PE s = 1 2 k x2, then W s = ( PE sf PE si ) W s = PE s x 0 = 0 W s =( 1 2 k x f W s = ( 1 2 k x f x i d x f 2 ) ( 1 2 k x 2 i ) k x 2 i ) > Elastic work is just the negative of the change in elastic potential energy. + This is similar to the gravitational potential energy case. + It is also path-independent. + Take care when assigning the reference point for the elastic case. > Note that only conservative forces lead to path-independent work. > Only conservative forces yield potential energies. 8

9 Conservation of Energy > Recall the work-energy theorem: W tot = KE W NC +W C = KE where the nonconservative & conservative parts of work are shown. W NC PE= KE W NC = KE+ PE W NC =(KE f KE i )+(PE f PE i ) W NC =(KE f + PE f ) ( KE i + PE i ) Let KE + PE = E, then W NC = E f E i W NC = E > E represents the total mechanical energy of the system. + It changes due to nonconservative work. + Forces that dissipate energy decrease the energy of a system. > If nonconservative effects are negligible, then the energy remains constant. E=0 E f = E i KE f +PE f = KE i + PE i > The kinetic & potential energies may change, but their sum remains the same in any state. 9

10 Power > Power describes how fast energy is transferred. + Electrical output, biological activity + Energy transfer per unit time: Unit of Power: watt (W) Work done Ave. Power= Duration of Work done P= W t 1 W = 1 J/s = 1 kg-m 2 /s 3 1 horsepower (hp) = 550 ft-lb/s = 746 W > Power can also be written in terms of the force acting on a moving object: P= W t = F x t P=F v Kilowatt-hour (kwh): amount of energy used in 1 h at a rate of constant rate of 1 kw 1 kwh = (1000 W) x (3600 s) = 3.6 x 10 6 J 10

11 Summary > Potential energy refers to the energy due to a body's position or state; it is related to a conservative force acting on a body. PE g =m g y PE s = 1 2 k x2 > The mechanical energy of a system is conserved if only conservative forces act on a system. E f = E i KE f +PE f = KE i + PE i > Power refers to the rate at which energy is transformed. P= W t 11

12 Sample Problems (Serway, Giancoli, etc.) 1. A 75.0-kg skier rides a 2830-m-long lift to the top of a mountain. The lift makes an angle of 14.6º with the horizontal. What is the change in the skiers gravitational potential energy? (Cutnell) 12

13 Sample Problems (Serway, Giancoli, etc.) 2. A diver of mass m drops from a board 10.0 m above the water's surface... Neglect air resistance. (a) Use conservation of mechanical energy to find his speed 5.00 m above the water's surface. (b) Find his speed as he hits the water. (Serway) 13

14 Sample Problems (Serway, Giancoli, etc.) 3. You are trying to lose weight by working out on a rowing machine. Each time you pull the rowing bar (which simulates the oars ) toward you, it moves a distance of 1.2 m in a time of 1.5 s. The readout on the display indicates that the average power you are producing is 82 W. What is the magnitude of the force that you exert on the handle? (Cutnell) 14