Lecture 3. > Potential Energy. > Conservation of Energy. > Power. (Source: Serway; Giancoli) Villacorta--DLSUM-SCIENVP-L Term01

Transcription

1 Lecture 3 > Potential Energy > Conservation of Energy > Power (Source: Serway; Giancoli) 1

2 Conservative & Nonconservative Forces > 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.

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 Reference pt. (Pe g = 0; y = 0) 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 F= 1 k x = k x+0 6

7 Elastic Potential Energy contd > Thus, W s =( 1 k x) x > Consider the work done by a spring stretched from x i to x f : F= k ( x f + x i ) W s = 1 k ( x) W s = 1 k x > 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 k ( x f + x i )] ( x f x i ) x i d W s = 1 k ( x f x i ) x f 7

8 Elastic Potential Energy contd F= k ( x f + x i ) Let PE s = 1 k x, then W s = ( PE sf PE si ) W s = PE s x 0 = 0 W s =( 1 k x f W s = ( 1 k x f x i d x f ) ( 1 k x i ) 1 k x 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 /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 k x > 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 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) 1

13 Sample Problems (Serway, Giancoli, etc.). While running, a person dissipates about 0.60 J of mechanical energy per step per kilogram of body mass. If a 60-kg person develops a power of 70 W during a race, how fast is the person running? (Assume a running step is 1.5 m long.) (Serway) 13