Contents. Objectives Work II Potential Energy Energy Conservation Potential Wells Bar Graphs Recap. Contents

Transcription

1 Physics 121 for Majors Class 15 Work, nergy, Potential nergy Conservation of Mechanical nergy Last Class Test Review Dot Products Work, force, energy Today s Class More on Work Potential nergy Conservation of nergy Section 1 Work II A mass m slides without friction down the plane. How much work does the normal force do? 1

2 A mass m slides without friction down the plane. How much work does the gravitational force do? A mass m slides without friction down the plane. How much work does the gravitational force do? sin It doesn t depend on θ! If the block slides down the plane without friction,. What is the work done by gravity if there is a frictional force? If the block slides down the plane without friction,. What is the work done by gravity if I pull the block down the plane with a rope? If the block slides down the plane without friction,. What is the work done by gravity if I pull the block up the plane with a rope? Double Plane A block slides frictionlessly down a plane that has two different angles. How much work does gravity do this time? 2

3 Conclusions Without friction, the work done by gravity is always the same. Dragging Masses You pull a series of blocks a distance d along a frictionless surface with an ideal rope of tension T. ach block has mass m If gravity is the only force other than the normal force, the total work done is the work done by gravity. The velocity of the block is always the same at the bottom of the incline. What is the tension in each rope? How much total work is done on each mass? How much work is done on each rope? Atwood s Machine What is the tension in the rope? How much work is done my the rope on each mass in moving a distance? How much work is done by gravity in moving a distance? How much total work is done on the two-mass system? What is the total change in kinetic energy of the two-mass system? Section 2 Potential nergy Definition of nergy from the Greek for activity, action, operation the ability to do work kinetic energy is ½ 2, a convenient measure of motion, but different than momentum anything that can be converted to kinetic energy nergy in Relativity In relativity, energy is 2 where is the speed of light and is mass. Mass, as determined by collisions, is measured to be larger when an observer measures the speed to be larger. When I see 1 kg of nails going by on a truck, I say its mass is larger than 1 kg. The driver says it is just 1 kg. So ( total energy ) includes both mass and kinetic energy. 3

4 nergy in Relativity 2 is conserved in all interactions We say momentum carries a particle forward in space Conservation of nergy More generally we say energy can take on many different forms nergy is always conserved if we look at all the possible forms of energy We say energy carries a particle forward in time We often consider only mechanical energy. Then energy is not always conserved. Conservation of nergy First proposed by Julius Robert von Meyer in Potential nergy and Work The work I do to a system represents its change in total energy which is also its change in kinetic energy. When I raise an object off the table slowly, the net work done on the object may be (nearly) zero. But if I let go, gravity does work on the object, increasing its kinetic energy. It is useful to invent something called potential energy that I can think of as flowing into an object as I do work on it against the force of gravity. It has the potential to become kinetic energy. Internal nergy If I do work against frictional force (I push one way and friction pushes the other), and let an object go, the friction doesn t push the object back where it came from. Friction doesn t make potential energy. Internal nergy We say work done against friction increases internal energy internal energy includes heat or any other energy that can t turn into bulk motion. Where did the work go? 4

5 Potential nergy If we do work against a force and the amount of kinetic energy that results equals the work we ve done, we say we have put potential energy in a system. Potential energy isn t something real - we can t suck it out of a book after we raise it. But potential energy is very useful. Gravitational Potential nergy A mass m slides without friction down the plane. How much work does the gravitational force do? sin It doesn t depend on θ! Gravitational Potential nergy The work done by gravity when I raise an object a distance is. Gravity will do when the object falls back again. This is the gravitational potential energy. The formula only defines because that s all we can measure. Potential nergy Formulas Gravity We often say. But where is 0? Potential nergy Formulas Astronomical Gravity The minus sign means the force is attractive. We sometimes ignore it, but we ll use it here. Potential nergy Formulas Spring 1 2 The equilibrium (unstretched position) of the spring is. We choose 0. We choose 0 so We often choose 0as well. 5

6 Section 3 Conservation of Mechanical nergy Conservation of nergy The work energy theorem says If there is a potential energy Then 0 Or where is the total mechanical energy, not the total relativistic energy! Conservation of nergy Bottom line: If we know and at one point in space and we know a formula for everywhere: 1. is a function of position only! 2. We can find everywhere! two eamples Penny and mpire State Building If there were no drag forces, how fast would a coin dropped from the mpire State building go when it hits the ground? Its height is 381 m. (86.4 m/s = 193 mph) Spring Gun A toy gun is made by compressing a spring and shooting a steel ball of mass 10 g. The spring compresses 3.20 cm and has a constant of k = 126 N/m. What is the force on the ball? What is the muzzle velocity of the gun? (4.03 N, 3.59 m/s) 6

7 Section 4 Potential nergy Wells Potential nergy Graph We often call this the potential enrgy well. We ll only consider one dimentsion for now. Potential nergy Graph What does this graph tell you? () Potential nergy Graph Now we ll add a line for total energy. () Where is the potential energy a minimum? Where is the kinetic energy a minimum? Where is the kinetic energy a maimum? Why is this line a straight line? What is the minimum kinetic energy? What is the maimum kinetic energy? Potential nergy Graph How we add kinetic energy to the graph? () Potential nergy Graph What is the direction of the force at each? 5 1 ()

8 Potential nergy Graph Where is the force the largest? How can you tell? () Potential nergy and Force Important result!!! now this!!! In one dimension: This is because or or Potential nergy and Force In three dimensions where is the gradient operator. This means Potential nergy and Force Let s say,, Then 2 2 You won t need to use this, but Now, back to the graph Potential nergy Graph What happens when put an object at a certain and let it go from rest? What is? () Potential nergy Graph How does it change if I add a constant to everywhere? () 2 2 Does it matter where we choose =0? 8

9 Section 5 Spring nergy Relaed no energy nergy Bar Graphs 10 Spring nergy Start compressing do work to the system Spring nergy Finish compressing Spring nergy Spring relaing Spring nergy Spring relaed

10 Spring nergy Spring stretching Section 6 10 Recap Big Ideas When systems have no work done to them when an object moves from one point back to the same point, we can predict the kinetic energy simply by knowing the object s position. In this case we can define a potential energy and say that mechanical energy (+) is conserved. Schedule Do Post-Class Quiz #15 Do Pre-Class Quiz #16 HW #14 is due Friday HW #15 is due Wednesday Quiz #4 is due Saturday (HW 12-13) Midterm #2 is on Tuesday, so do the Basic Test!!! 10