Chapter 6 Work and Energy
The Ideal Spring HOOKE S LAW: RESTORING FORCE OF AN IDEAL SPRING The restoring orce on an ideal spring is F x = k x SI unit or k: N/m
The Ideal Spring Example: A Tire Pressure Gauge The spring constant o the spring is 30 N/m and the bar indicator extends.0 cm. What orce does the air in the tire apply to the spring?
The Ideal Spring F Applied x = = k x ( 30 N m)( 0.00 m) = 6.4 N
Potential energy o an ideal spring W elastic = ( F cosθ)s = 1 ( kx + kx o )cos0 ( x o x ) F 0 F F = kx W elastic = 1 1 kx o kx x x 0 F = 1 ( F + F ) 0 s F
Potential energy o an ideal spring DEFINITION OF ELASTIC POTENTIAL ENERGY The elastic potential energy is the energy that a spring has by virtue o being stretched or compressed. For an ideal spring, the elastic potential energy is PEelastic = 1 kx SI Unit o Elastic Potential Energy: joule (J)
Conservative Versus Nonconservative Forces DEFINITION OF A CONSERVATIVE FORCE Version 1 A orce is conservative when the work it does on a moving object is independent o the path between the object s initial and inal positions. Version A orce is conservative when it does no work on an object moving around a closed path, starting and inishing at the same point.
Conservative Versus Nonconservative Forces
Conservative Versus Nonconservative Forces Version 1 A orce is conservative when the work it does on a moving object is independent o the path between the object s initial and inal positions. W gravity = mg( h o h )
Conservative Versus Nonconservative Forces Version A orce is conservative when it does no work on an object moving around a closed path, starting and inishing at the same point. W gravity = mg h o h ( ) h o = h è W gravity = 0
Conservative Versus Nonconservative Forces An example o a nonconservative orce is the kinetic rictional orce. W = F cosθ ( )s = k cos180 s = k s The work done by the kinetic rictional orce is always negative. Thus, it is impossible or the work it does on an object that moves around a closed path to be zero. The concept o potential energy is not deined or a nonconservative orce.
Conservative Versus Nonconservative Forces In normal situations both conservative and nonconservative orces act simultaneously on an object, so the work done by the net external orce can be written as W = W c + W nc W = KE KE o = ΔKE W c = Wgravity = mgho mgh = PE o PE = ΔPE
Conservative Versus Nonconservative Forces W = W c + W nc ΔKE = ΔPE +W nc A dierent version o the WORK-ENERGY THEOREM W nc = ΔKE + ΔPE
The Conservation o Mechanical Energy W nc = ΔKE + ΔPE = ( KE KE ) o + ( PE PE ) o W nc = KE + PE _ ( ) + ( KE o + PE ) o W nc = E E o = ΔE E = KE + PE è the total mechanical energy I the net work on an object by nonconservative orces is zero, then its energy does not change: E = E o
The Conservation o Mechanical Energy THE PRINCIPLE OF CONSERVATION OF MECHANICAL ENERGY The total mechanical energy (E = KE + PE) o an object remains constant as the object moves, provided that the net work done by nonconservative orces is zero.
The Conservation o Mechanical Energy
The Conservation o Mechanical Energy Example: A Daredevil Motorcyclist A motorcyclist is trying to leap across the canyon by driving horizontally o a cli 38.0 m/s. Ignoring air resistance, ind the speed with which the cycle strikes the ground on the other side.
The Conservation o Mechanical Energy 1 E = E o mgh + mv = mgh + o 1 mv o gh + v = gh + 1 o 1 v o
The Conservation o Mechanical Energy gh + v = gh + 1 o 1 v o v = g( h o h ) + v o v = 9.8m s ( ) 35.0m ( ) + 38.0m s ( ) = 46.m s
The Conservation o Mechanical Energy Conceptual Example: The Favorite Swimming Hole The person starts rom rest, with the rope held in the horizontal position, swings downward, and then lets go o the rope. Three orces act on him: his weight, the tension in the rope, and the orce o air resistance. Can the principle o conservation o energy be used to calculate his inal speed?
The Conservation o Mechanical Energy Example: Vertical spring with a mass A 0.0-kg ball is attached to a vertical spring. The spring constant is 8 N/m. When released rom rest, how ar does the ball all beore being brought to a momentary stop by the spring?
The Conservation o Mechanical Energy E = E 0 1 mv + mgh + 1 k(δy ) = 1 mv 0 + mgh 0 + 1 k(δy 0 ) v 0 = v = 0 Δy 0 = 0, since spring initially unstretched Δy = h 0 h = 0 1 kh 0 = mgh 0 h 0 = mg k = ( ) 9.8m s 0.0 kg ( ) 8N m = 0.14 m
Nonconservative Forces and the Work-Energy Theorem Example o a nonconservative orce problem: Fireworks A 0.0 kg rocket in a ireworks display is launched rom rest and ollows an erratic light path to reach the point P, as in the igure. P is 9 m above the starting point. In the process, 45 J o work is done on the rocket by the nonconservative orce generated by the burning propellant. Ignoring air resistance and the mass loss due to the burning propellant, ind the speed v o the rocket at point P. THE WORK-ENERGY THEOREM W nc = E E o W nc = ( mgh + 1 mv ) ( mgh o + 1 mv ) o
Nonconservative Forces and the Work-Energy Theorem W nc = mgh mgh o + 1 mv 1 mv o W nc = mg( h h ) o + 1 mv 45 J = ( 0.0 kg) ( 9.80m s )( 9.0 m) + 1 ( 0.0 kg)v v = 61m s
Power DEFINITION OF AVERAGE POWER Average power is the rate at which work is done, and it is obtained by dividing the work by the time required to perorm the work. Work P = = Time W t joule s = watt (W)
Power From the work-energy theorem, doing work results in a change in energy or the system, so alternatively: P = Change in energy Time 1 horsepower = 550 oot pounds second = 745.7 watts Another way to express the power generated by a orce acting on an object moving at some average speed, P = Fv