WEBVTT physics/ap-physics-c-mechanics/fullerton
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Hello, everyone, and welcome back to www.educator.com.
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I'm Dan Fullerton and in this lesson we are going to talk about Energy and Conservative forces.
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Our objectives include defining energy, describing various types of energy,
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talking about some alternative definitions of conservative force,
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describing some examples of conservative and non conservative forces.
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Finally using a relationship between force and potential energy to find forces and potential energy functions.
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Let us dive right in by talking about what is energy.
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Energy is the ability or capacity to do work.
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Work is the process of moving object.
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If we put those together energy is the ability or capacity to move on the object.
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Energy can be converted to different types.
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It can be transformed from one type to another.
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You see that all the time.
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You transfer energy from one object to another by doing work.
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We talked about the work energy theorem in our last lesson.
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The work done on a system by an external force changes the energy of that system.
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We are going to be talking about lots of energy transformations in the next couple lessons
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but one of those is going to come up again and again is kinetic energy.
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Kinetic energy is energy of motion or if energy is the ability or capacity to move an object.
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Kinetic energy that is the ability or capacity of a moving object to move another object.
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Something that is moving has the ability to move something else.
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The baseball coming toward your nose with a bunch of Velocity has kinetic energy.
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It has kinetic energy because it has the ability to move something else.
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Basically the bones in your nose.
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If that hits your nose it is going to squish it up.
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It is going to cause something else to move.
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Kind of a messy example but that is the idea.
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Kinetic energy is the capacity or ability of that moving object to move something else.
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The formula for kinetic energy is ½ mass × the square of your speed.
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Units of course energy Joules.
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Let us take a look at a quick example here.
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A frog speeds along a frog o cycle at a constant 30 m/s.
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If the mass of the frog and motorcycle is 5 kg find the kinetic energy of the frog cycle system.
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Kinetic energy is ½ MV² to be ½ × the mass 5 kg × our speed 30 m/s² 900 × 5=4500.
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½ of that is going to be 2250 joules.
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Let us talk now about potential energy.
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Potential energy which gets the symbol capital U is energy an object possesses based on its position or its speed of being.
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There are lots of different types of potential energy.
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Gravitational potential energy will talk about quite a bit in this course.
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It is the energy an object possesses because of its position in the gravitational field.
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Elastic potential energy we talked about a little bit with Hooke's law already.
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The energy that you have from some sort of elastic displacement something like a spring or an elastic band.
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Chemical potential energy, electric potential energy, nuclear potential energy, all different forms of potential energy.
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It is important to point out that a single object in isolation can only have kinetic energy.
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In order to have some from a potential energy you need to have an interaction between objects.
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You need to have at least two objects in your system or something they have potential energy.
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As we talk about these we are also going to run across the term internal energy every now and then.
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The internal energy of the system includes the kinetic energy of the objects that make up that system
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and potential energy of the configuration of the objects that make up the system.
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For example if you think of the temperature, the heat and temperature of an object that
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is based on the speed with which the molecules inside are moving around.
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That is an internal energy because it includes the kinetic energy of the objects comprising that system.
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We can make changes internally that system.
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The changed the internal energy of the object.
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Change in a systems internal structure can result in changes in the internal energy.
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Let us see if we can put some of us together as we talk about different types of energy.
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I like to break energy up in the two main types potential and kinetic.
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Some of these cross the boundary between each one.
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If we talk about electrical energy depending on how you are looking at moving charges create current.
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Moving there is kinetic energy.
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Light moving photons as they do not have mass but they do have energy.
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Light is another goofy one.
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We could talk about is kinetic energy.
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Wind is moving air molecules.
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Thermal energy the motion of molecules and atoms making up of an object.
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Sound vibrating air or vibrating molecules and waves.
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Potential side we have chemical potential energy, gravitational potential energy, we can talk about electrical potential energy,
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in terms of voltage charges held at different levels.
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Nuclear potential energy, elastic potential energy, and tons of others.
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Let us start with gravitational potential energy Ug.
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If we have some object we will start here at some arbitrary point we will call y = 0
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and we take it across a meandering path until eventually gets to another position.
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If this is our +y direction well we have changed its gravitational potential energy.
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How much if we change it?
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Well let us take a look.
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The work done in moving it there which should be the amount of potential energy that has its final point.
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As we are dealing with the conservative force we will talk about that in a minute.
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As we go from y=0 to Y equals some final value let us call that the h level.
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So that is h.
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F • dr define their work which would be the integral from 0 to h are force.
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To lift that up we have to overcome the force of gravity so that is going to be MG and DR.
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We are worried about the Dy position.
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MG should be constant as long as we are in the same relative level, the same gravitational field if we are going miles and miles up.
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A very big distance into the atmosphere G might change.
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As long as we are relatively close to use pretty constant.
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We will pull those out MG we should say work equals MG integral from 0 to H.
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Dy which is just going to be MGH.
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As long as are the constant gravitational field gravitational potential energy can be written as MGH you have probably seen that before.
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Let us take a look and example here.
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In the diagram we have 155 N box on a ramp.
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Applied force F causes the box to slide from point A to point B.
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What is the total amount of gravitational potential energy gained by the box?
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As I look at this change in gravitational potential energy is going to be MG Δ h or change in height.
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Which is going to be 155 N that describes its weight or the force of gravity MG on it.
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Δ H and goes up 1.8 m.
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That would be 279 joules pretty straightforward.
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Let us take a look at graph in question.
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The hippopotamus is throwing vertically upward.
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I do not know why and I really do not know how.
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Which pair of graphs best represent the hippos kinetic energy in gravitational potential energy as functions of its displacement while it rises?
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As it goes higher and higher as it rises, it is going to slow down so we have to expect that we are going to see kinetic energy decreasing.
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On the other hand as it gets higher and higher its gravitational potential energy should be increasing.
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The kinetic energy is being transformed into gravitational potential energy.
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Which graph we have that displays that the best?
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That looks like one as we get further and further displacements on its way up kinetic energy goes to 0 at its highest point where it stops.
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That is where we have our maximum of gravitational potential energy.
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The answer must be 1.
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Looking at a slightly more involved question.
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A pendulum of mass M swings on the light string of length L.
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If the mass hanging directly down is set to 0. Of gravitational potential energy
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find a gravitational potential energy the pendulum as a function of θ and L.
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It looks like if I break this up a little bit or really doing is we are changing the height of our pendulum as it swings from there to there.
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There is our change the Δ Y.
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We have to figure out what that is.
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To do that I am going to take a look at my triangle here and realize that our hypotenuse here is going to be equaled L.
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We got the adjacent side of our angle right here and this would be the opposite side.
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To help figure out that height let us say that the cos θ SOHCAHTOA.
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Cos is adjacent over hypotenuse.
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That is a adjacent over hypotenuse which implies that this adjacent side, this piece right here from there to there is going to be the hypotenuse × cos θ.
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Since our hypotenuse is the length of our string L that is going to be L cos θ.
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Our Δ y is going to be the entire length L - adjacent side which is L cos θ.
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I'm going to find that Δ y is going to be L × (1 - cos θ).
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If I wanted a change in potential energy I can write Δ Ug = Mg Δ Y.
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Which would be Mg × L 1 - cos θ.
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We have our gravitational potential energy as a function of θ and L for our swinging pendulum.
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Let us talk for a minute about conservative forces.
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A force in which the working on an object is independent of the path is known as a conservative force.
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Gravity for example.
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If I left a pen straight up or go to the side and run it around all over the place, the only thing that matters
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for the total change in the energy or the total work done is the initial and final points.
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You can also write that define a conservative forces of force in which the work done moving along the close path is 0.
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As long as you come back to where you start whoever you get there and net work done must be 0.
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Or it is a force in which the work that is directly related to a negative change in the potential energy.
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The work that is equal to the opposite of the change in the potential energy.
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Some examples of these.
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Let us talk about some conservative forces and some non conservative forces.
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Conservative forces would be things like gravity.
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We are only worry about the initial and final points or elastic forces.
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When we talk about electricity in the following course on ENM columbic or electric force.
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Non conservative forces are one you typically think of as the law C forces for example.
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Friction the energy is ever destroy your loss that can be converted less useful forms.
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Friction drag forces, retarding forces, air resistance, fluid resistance, non conservative forces.
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There are some properties of conservative forces.
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For the work done by conservative forces, let us say it is a work done by a conservative force
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is equal to the opposite of the change in its potential energy.
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Which implies then that Δ U equals the opposite of the work done by the conservative force.
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Which implies that the Δ U equals the opposite of the integral from some initial position to some final position of F•Dr.
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And we can use this in a bunch of different ways.
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We are going to apply it with a couple different types of conservative forces in the next few slides.
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Let us take a look at in the context of newton's law the universal gravitation.
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You have probably seen this before and other courses.
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I do not know if it is formally introduced here yet but the gravitational force between two objects is - G universal gravitational constant.
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It is a fudge factor to make the unit is work out equal to 6.67 × 10⁻¹¹ Nm² / kg².
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A constant × first mass × second mass divided by the square of the distance between their centers.
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The r hat just tells you the direction of r hat in the direction of the unit vector from the first object
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to the second in the negative tells you that it is going to be attractive there.
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We can look at it as here we have mass 1.
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Somewhere over here we have a mass 2.
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The distance between their centers we can draw that right in nice and quick.
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There it is that would be our r vector.
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R hat would just be a unit vector in that direction.
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If we start with a Newton’s law of universal gravitation we can use what we just found out about conservative forces to find a potential energy.
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The universal gravitational potential energy.
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Change in potential energy is minus the work done by conservative forces is going to be - the work done by gravity
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or - the integral as we go from infinity to r some point of - GM 1 M2 /r² with respect to r.
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As I do this integration I can pull my constant out.
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Potential energy due to gravity is going to be GM 1 M 2 should not change for the purposes of this problem.
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This will be GM1 M2 our negative and negative cancel out.
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Integral from infinity to some r of DR/ r² or r⁻² Dr.
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Let us write it that way just to make it a little easier to see.
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GM 1 M2 integral from infinity to r of r⁻² DR is going to be GM 1M2.
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The integral of r⁻² is going to be 1/r.
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Evaluated from infinity to r so potential energy due to gravity is just going to be GM 1M2 with 1.
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We will plug that in - GM 1M21 1/ infinity will be 0.
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I'm going to come up with - GM 1 M2 /r.
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And that is our formula for gravitational potential energy between two objects separated by some distance r.
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We are able to derive it knowing the force because it is a conservative force.
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We can do the same thing with the elastic potential energy.
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Remember the force in the spring by Hooke’s law – KX.
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The work done is negative change in potential energy and that is a conservative force or - the integral from 0 to X of F• DR
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as we are finding the work done which is - the integral from 0 to X of – KXDX.
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The potential energy stored in the spring is the integral from 0 to X of KXDX which is K plot a constant integral from 0 to X.
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0 to X of XDX or K × X²/2 evaluate from 0 to X which is just going to end up being ½ KX 2.
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There is a potential energy stored in the spring.
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If we can go from force to potential energy we should be able to get force from potential energy.
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Let us take a look at that.
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As soon as we have an object on some path dr or we could also call this DL that might be just as common dl.
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As it travels along this path some forces acting on a net force can be changing.
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If we want to find V potential energy to this force along the path what we can do is break it up
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and these little tiny pieces DL and find the potential energy for each one of those by finding the work done through each one of those.
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The differential potential energy that tiny amount of potential energy is the opposite of that little tiny bit of work done by the conservative force.
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Or - F dotted with will use dl there which if we want to force in the direction DL this is going to be - F cos θ DL.
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If you remember our dot product definition.
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What we are going to do is we are going to call this F cos θ we are going to call that the force in the direction of L.
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We will call that fl.
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Once we define that the little tiny bit of energy, potential energy is - F cos θ DL which we could also write as – FLDL.
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That means that FL the force in the direction of that displacement is just going to be - du DL.
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The opposite of the derivative of the potential energy function with respect to L gives you that force.
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We can go from potential energy to force.
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Let us do that with gravity.
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FL equals - du DL and their potential energy function for gravity is - GM 1M2/r.
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Let us say we are looking for force in the direction r.
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Force r equals - the derivative with respect to R of our potential energy function - GM 1M2/r.
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We can pull out our constants GM 1M2 × the derivative with respect to r of 1/r
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which means that the force in the direction of R is going to be GM 1M2.
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The derivative of 1/ R is -1/ r² or - GM 1M2 /r² which is what we had initially force = - GM 1M2/r² in that direction of r hat.
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We have gone from force to potential energy.
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We can go back the other way using this formula.
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Let us take a look at Hooke’s law.
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Here we are going to take a look at a spring and we are going to give its potential energy function as ½ KX².
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As you extend it to the right you have more potential energy.
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The area under the graph from Ug and no kinetic.
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If you are let it go with the object, it gets to its center pointed it no longer has elastic kinetic energy instead it so elastic potential energy.
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It is all kinetic.
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We get to the other side it slows down and stops with all potential again it stretch out or compressed spring.
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And then you come back.
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You have this constant back and forth potential kinetic energy.
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We can look at this in terms of the force realizing that - du DL =Fl.
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The force in the direction of the L is – d/dx for spring of ½ KX² which is just going to be – KX.
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We have gone the other way again.
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To put a quick summary together.
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If we start with potential energy something like gravitational potential energy is MGH.
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We can look at the force - Du DL.
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A force the derivative of this with respect to h is going to be – MG.
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There is our formula for weight.
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Assuming you are in a constant gravitational field.
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If you want to get more general with Newton’s universal gravitation, potential energy due to gravity is - GM 1M2/r.
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The force using - dudl is going to be - GM 1M2/r².
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Newton’s law of universal gravitation.
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Or if we start with the spring where the stored potential energy this spring is ½ KX² in a linear spring
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that follows Hooke’s law we can find the force by taking the opposite of the derivative that with respect to x which is – KX.
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We can go that direction.
00:24:33.800 --> 00:24:39.300
We previously went that direction.
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Let us do a couple examples.
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5 kg sphere’s position is given by the function x of T = 3T³ -2T.
00:24:48.600 --> 00:24:56.500
The term in the kinetic energy of the sphere at time T = 3s.
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Kinetic energy is ½ MV².
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It would be helpful to know the velocity.
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If we know x (t) is t3³- 2t the velocity is the first derivative of x which will be 9t² -2.
00:25:18.700 --> 00:25:28.200
The velocity of t = 3s is just going to be 9 × 9=81 – 2 = 79 m/s.
00:25:28.200 --> 00:25:31.100
We can go back to our kinetic energy function.
00:25:31.100 --> 00:25:47.000
Kinetic energy is ½ × 5kg × speed 79 m/s or about 15600 joules.
00:25:47.000 --> 00:25:50.400
An example where we find force from potential energy.
00:25:50.400 --> 00:25:59.200
Another 5kg sphere’s potential energy U is described by this function U(x) = 4x² + 3x – 2.
00:25:59.200 --> 00:26:03.700
Determine the force on the particle at x = 2m.
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Force = -du dl in this case it is going to be – d /dx of 4x² + 3x – 2= -derivative of 4x² 8x.
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The derivative of +3x is 3, -2 is 0.
00:26:26.900 --> 00:26:31.100
Which is -8x – 3.
00:26:31.100 --> 00:26:43.800
If we want the force at x = 2m that is going to be -8 × 2m -3.
00:26:43.800 --> 00:26:54.500
-16 – 3 = -19 N.
00:26:54.500 --> 00:26:56.400
One last problem here.
00:26:56.400 --> 00:26:58.500
Work on a spinning disc.
00:26:58.500 --> 00:27:04.400
A 2kg disc moves in uniform circular motion on a frictionless horizontal table.
00:27:04.400 --> 00:27:11.400
Attach to the point of rotation by a 10cm spring but the spring constant of 50 N/m.
00:27:11.400 --> 00:27:14.200
When stationary the spring has a length of 8cm.
00:27:14.200 --> 00:27:18.000
While it is turning it is extended 2cm.
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How much work is performed on the disc by the spring as the disc moves through 1 full revolution?
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The force in order to have it moving in a circle must be that way.
00:27:30.800 --> 00:27:34.400
The velocity at any given point in time is that way.
00:27:34.400 --> 00:27:37.500
They are perpendicular.
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If we do not have any force in the direction of the displacement you cannot do any work.
00:27:42.900 --> 00:27:45.900
F cos θ cos 90 is 0°.
00:27:45.900 --> 00:27:53.100
Therefore the work done is 0.
00:27:53.100 --> 00:27:55.300
A tricky question there.
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Hopefully that gets you a good start on energy and conservative forces.
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Thank you so much for joining us here at www.educator.com.
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I hope to see you again real soon.
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Make it a great day everybody.