# Work and Energy continued

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1 Chapter 6 Work and Energy continued

2 Requested Seat reassignments (Sec. 1) Gram J14 Weber C22 Hardecki B5 Pilallis B18 Murray B19 White B20 Ogden C1 Phan C2 Vites C3 Mccrate C4

3 Demonstrations Swinging mass, pendulum, collisions Ramps and slides Spinning mass inside toy truck

4 Clicker Question 6.2 Two balls of equal size are dropped from the same height from the roof of a building. One ball has twice the mass of the other. When the balls reach the ground, how do the kinetic energies of the two balls compare (ignoring air friction)? a) The lighter one has one fourth as much kinetic energy as the other does. b) The lighter one has one half as much kinetic energy as the other does. c) The lighter one has the same kinetic energy as the other does. d) The lighter one has twice as much kinetic energy as the other does. e) The lighter one has four times as much kinetic energy as the other does. Hint : what is the acceleration and v f of each mass?

5 Clicker Question 6.2 Two balls of equal size are dropped from the same height from the roof of a building. One ball has twice the mass of the other. When the balls reach the ground, how do the kinetic energies of the two balls compare? a) The lighter one has one fourth as much kinetic energy as the other does. b) The lighter one has one half as much kinetic energy as the other does. c) The lighter one has the same kinetic energy as the other does. d) The lighter one has twice as much kinetic energy as the other does. e) The lighter one has four times as much kinetic energy as the other does. Acceleration (g), and v f are the same for both masses. KE = 1 2 mv f2, proportional to mass. KE 1 = 1 2 m 1 v f2 ; KE 2 = 1 m v 2 = 1 (2m )v 2 2 f 2 1 f2 = 2 1 m v f = 2KE 1 KE 1 = 1 2 KE 2

6 6.3 Gravitational Potential Energy GRAVITATIONAL POTENTIAL ENERGY Energy of mass m due to its position relative to the surface of the earth. Position measured by the height h of mass relative to an arbitrary zero level: PE replaces Work by gravity in the Work-Energy Theorem Work-Energy Theorem becomes Mechanical Energy Conservation: KE f + PE f = KE 0 + PE 0 E f = E 0 Initial total energy, E 0 = KE 0 + PE 0 doesn't change. It is the same as final total energy, E f = KE f + PE f. Another way (equivalent) to look at Mechanical Energy Conservation: ( KE f KE ) 0 + ( PE f PE ) 0 = 0 ΔKE + ΔPE = 0 Any increase (decrease) in KE is balanced by a decrease (increase) in PE. These are used if only Conservative Forces act on the mass. (Gravity, Ideal Springs, Electric forces)

7 6.5 The Conservation of Mechanical Energy Sliding without friction: only gravity does work. Normal force of ice is always perpendicular to displacements.

8 Clicker Question 6.3 You are investigating the safety of a playground slide. You are interested in finding out what the maximum speed will be of children sliding on it when the conditions make it very slippery (assume frictionless). The height of the slide is 2.5 m. What is that maximum speed of a child if she starts from rest at the top? a)1.9 m/s b) 2.5 m/s c) 4.9 m/s d) 7.0 m/s e) 9.8 m/s

9 Clicker Question 6.3 You are investigating the safety of a playground slide. You are interested in finding out what the maximum speed will be of children sliding on it when the conditions make it very slippery (assume frictionless). The height of the slide is 2.5 m. What is that maximum speed of a child if she starts from rest at the top? a)1.9 m/s b) 2.5 m/s c) 4.9 m/s d) 7.0 m/s e) 9.8 m/s E 0 = mgh; E f = 1 mv 2 2 f E 0 = E f mgh = 1 mv 2 2 f v f = 2gh = 2(9.8)(2.5) m/s =7.0 m/s

10 6.4 Conservative Versus Nonconservative Forces In many situations both conservative and non-conservative forces act simultaneously on an object, so the work done by the net external force can be written as W Net = W C +W NC W C = work by conservative force such as work by gravity W G Work-Energy Theorem becomes: KE f + PE f = KE 0 + PE 0 +W NC But replacing W C with (PE f PE 0 ) work by non-conservative forces will add or remove energy from the mass E f = E 0 + W NC E f = KE f + PE f E 0 = KE 0 + PE 0 Another (equivalent) way to think about it: ( KE f KE ) 0 + ( PE f PE ) 0 = W NC ΔKE + ΔPE = W NC if non-conservative forces do work on the mass, energy changes will not sum to zero

11 6.5 The Conservation of Mechanical Energy But all you need is this: KE f + PE f = KE 0 + PE 0 +W NC non-conservative forces add or remove energy If W NC 0, then E f E 0 If the net work on a mass by non-conservative forces is zero, then its total energy does not change: If W NC = 0, then E f = E 0 KE f + PE f = KE 0 + PE 0

12 Clicker Question 6.3 A mass has a total energy of 100 J. Then a kinetic frictional force does 50 J of work on the mass. What is the resulting total energy of the mass? a) 0J b) 50J c) 100J d) 150J e) 50J

13 Clicker Question 6.3 A mass has a total energy of 100 J. Then a non-conservative force does 50 J of work on the mass. What is the resulting total energy of the mass? a) 0J b) 50J c) 100J d) 150J e) 50J E f E 0 = W NC E f = E 0 +W NC = 100J + ( 50J) = 50J

14 6.5 The Conservation of Mechanical Energy Example 8 A Daredevil Motorcyclist A motorcyclist is trying to leap across the canyon by driving horizontally off a cliff 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side.

15 6.5 The Conservation of Mechanical Energy mgh f + 1 mv 2 = mgh 2 f mv gh f + 1 v 2 = gh 2 f v 2 2 0

16 6.5 The Conservation of Mechanical Energy gh f + 1 v 2 = gh 2 f v v f = 2g( h 0 h ) 2 f + v 0 v f = 2( 9.8m s 2 )( 35.0m) + ( 38.0m s) 2 = 46.2m s

17 6.5 The Conservation of Mechanical Energy Conceptual Example 9 The Favorite Swimming Hole The person starts from rest, with the rope held in the horizontal position, swings downward, and then lets go of the rope, with no air resistance. Two forces act on him: gravity and the tension in the rope. Note: tension in rope is always perpendicular to displacement, and so, does no work on the mass. The principle of conservation of energy can be used to calculate his final speed.

18 6.6 Nonconservative Forces and the Work-Energy Theorem Example 11 Fireworks Assuming that the nonconservative force generated by the burning propellant does 425 J of work, what is the final speed of the rocket (m = 0.2kg). Ignore air resistance. E f = E 0 +W NC ( ) ( mgh mv 2 ) 2 0 W NC = mgh f + 1 mv 2 2 f ( ) mv f2 + 0 = mg h f h 0 ( ) v f 2 = 2W NC m 2g h f h 0 = 2( 425 J) / ( 0.20 kg) 2( 9.80m s 2 )( 29.0 m) v f = 61 m/s

19 6.7 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 perform the work. P = Work Time = W t Note: 1 horsepower = watts P = W t = Fs t = F s t = Fv

20 6.7 Power

21 6.8 Other Forms of Energy and the Conservation of Energy THE PRINCIPLE OF CONSERVATION OF ENERGY Energy can neither be created not destroyed, but can only be converted from one form to another. Heat energy is the kinetic or vibrational energy of molecules. The result of a non-conservative force is often to remove mechanical energy and transform it into heat. Examples of heat generation: sliding friction, muscle forces.

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