AP Physics 1 review of Energy and Work

not only are there many different kinds of energies but both objects and systems of objects can have energy once they have that energy they can transfer it to another system or object or that energy could transform to a different type of energy inside that system when energy gets transferred we call that work and the amount of work that's done is the amount of energy that was transferred you often hear people say energy is conserved which really just means that you can't create or destroy energy you can simply transfer it between objects or systems so what are all the different types of energy there's kinetic energy which is the energy due to something moving and the formulas 1/2 the mass times the speed squared there's gravitational potential energy which is the energy something has due to its height and the formula is the mass times the magnitude of the acceleration due to gravity times the height of the object height above what height above whatever you're choosing is the H equals zero reference line is that cheating no because all that really matters is the change in gravitational potential energy not the actual value itself there's also spring potential energy which has to do with the compressed or stretched spring and the formulas 1/2 times the spring constant times X which is not the length of the spring X is the amount the spring has been compressed or stretched these three types of kinetic and potential energy constitute what we call mechanical energy mechanical energy is another word for the kinetic energy plus gravitational potential energy plus spring potential energy in a system and it's important to know that mechanical energy does not include thermal energy thermal energies the heat energy generated by dissipative forces like friction and air resistance and you can find the amount of thermal energy generated by taking the size of the dissipative force times the distance through which that force was acting the unit of energy is joules and energy is not a vector but maybe the most important thing to remember about energy is if there's no external work done on a system then there's no change in the energy of that system in other words if there's no external work done on a system the initial energy of that system will equal the final energy of that system which is the way you solve many conservation of energy problems so what's an example problem involving energy look like let's say a box started with an issue speed and slides from one platform up to another platform will assume that frictional forces and air resistance are negligible and for the system that's consisting of the mass and the earth what's happening to the total mechanical energy in this system so you got to pay special attention to what is in your system since my system includes the mass which is going to be moving my system is going to have kinetic energy and since my system has two objects that are interacting gravitationally the mass in the earth my system is also going to have gravitational potential energy so when I ask about the total mechanical energy of the system that's really just code for the total kinetic and potential energy of the system so as this mass slides up to a higher point on the ramp the gravitational potential energy increases but the mass is going to slow down so the kinetic energy is going to decrease however since the earth and the mass are in our system and there's no dissipative forces there's no external work done on our system yes the earth is doing work on the box but the earth is part of our system so it can't do external work and that means energy just gets transferred from one form to another within our system and the total mechanical energy here is going to remain the same for the entire trip now what if we ask the same question but we consider a system that consists only of the Box in that case our system has a box that's moving so it will have kinetic energy but our system no longer includes two objects interacting gravitationally so our system will have no gravitational potential energy what happens to the total mechanical energy in this case well the only energy that I've got in my system now is kinetic energy and since that kinetic energy decreased the total mechanical energy of the box as a system decreases how does it decrease it decreases because now the earth is outside of our system and the work that it is doing on the box is external work and it's taking away energy from the box what does work mean in physics work is the amount of energy transferred from one system or object to another in other words if a person lifted a box and gave it 10 joules of gravitational potential energy we'd say that that person did positive 10 joules of work on the box since that person gave the Box 10 joules of energy but since the box took 10 joules of energy from that person we'd say that the box did negative 10 joules of work on the person since the box took 10 joules of energy so you can find the work done if you can determine the amount of energy that was transferred but there's an alternative formula to find the work done if something's having work done to it there's got to be a force on that object and that object has to be displaced so if you take the force on the object times the displacement of the object and multiply by the cosine of the angle between the force and the displacement you'll also get the work done in other words one way to find the work done is by finding the amount of energy that was transferred but another way to find the work done is by taking the magnitude of the force exerted on an object times the displacement of the object and then times cosine of the angle between the displacement and the force since work is a transfer of energy it also has units of joules and even though work is not a vector it can be positive or negative if the force on an object has a component in the direction of motion that force will do positive work on the object and give the object energy if the force on the object has a component in the opposite direction of the motion the work done by that force would be negative and it would take away the object's energy and if the force on an object is perpendicular to the motion of the object that force does zero work on the object it neither gives the object energy nor takes away the object's energy so what's an example problem involving work look like let's say a box of mass M slides down a frictionless ramp of height H and angle 2 theta as seen in this diagram here and a separate box of mass 2 m slides down another frictionless ramp of height H and angle theta as seen in this diagram here and we want to know how the work done on the object by the earth compares for each case the easiest way to find the work done here is by finding the change in energy the box will gain an amount of kinetic energy equal to the amount of potential energy that it loses so the work done by the earth is just going to equal positive MGH both Heights are the same so the h's are equivalent but one box has twice the mass so the work done by gravity on the mass 2m is going to be twice as great as the work done on the mass one M what's the work energy principle mean the work energy principle states that the total work or the net work done on an object is going to be equal to the change in kinetic energy of that object so if you add up all the work done by all forces on an object that's got to be equal to the change in the kinetic energy that object in other words 1/2 MV final squared minus 1/2 MV initial squared so this is a really handy way to find how the speed of an object changes if you can determine the net work on an object in other words if there's multiple forces on an object and you can find the work done by each of those forces you can determine how much kinetic energy that object gained or lost so what's an example of the work-energy principle let's say a four kilogram box started with a velocity of 6 meters per second to the left some net amount of work is done on that box and it's now moving with a velocity of 4 meters per second to the right we want to know what was the amount of net work done on the box without even solving it we can say since this object slowed down energy was taken from it so the amount of network had to be negative which means it's either B or D to figure out which one exactly we could use the work-energy principle which says that the net work done is equal to the change in kinetic energy so if we take the final kinetic energy which is one-half times 4 kilograms times the final speed squared and we subtract the initial kinetic energy one-half times 4 kilograms times the initial speed squared 6 meters per second we get negative 40 joules of network if you get a force versus position graph the area under that graph will represent the work done so when you see F versus X you should think area equals work but be careful area above the x-axis is going to count as positive work done an area underneath the x-axis is going to count as a negative work done and make sure the x-axis really is position if you get a force versus time graph the area's impulse not work so what would an example of work as area look like let's say a box started at x equals zero with a velocity of five meters per second to the right and a net horizontal force on the box is given by the graph below we want to know at what position other than x equals zero will the box again have a velocity of five meters per second to the right well since the box will end with the same speed that it began with the changing kinetic energy is going to equal zero but that means the network would also equal zero since the network is equal to the change in kinetic energy so if the box starts at x equals zero how far do we have to go in order for us to have no net work done between zero and three meters the work done is going to be negative and the area of this triangle is going to be one half the base times the height which is one half times three meters times negative six Newton's which is negative nine joules of work done and the area under this triangle between three and five seconds would again be one half base times height which is one half times two meters times the height of four Newton's which is positive four joules of work done so by the time that the box is made it to five meters there's been a total amount of work done of negative nine plus four which is negative five joules of work but we want no net work done so we're going to have to keep going until this positive area contribution is going to equal the negative area in other words if I can make it so that all of this negative area is equal to all of the positive area my net work is going to equal zero my negative area is negative nine my positive area so far is positive four if I continue on to the six meter mark I'll pick up another positive four joules of work since the height of this rectangle is four and the width is one meter which means we're almost there four plus four is eight I'd only need to pick up one more Joule so I can't go all the way to seven meters I'd only need to go one more fourth of a meter to pick up one more Joule so that one plus four plus four is equal to negative nine so the net work would equal zero somewhere between x equals 6 and x equals seven which would ensure that the change in kinetic energy is zero and we would end with the same speed that we began with what does power mean in physics power is the amount of work done per time which can also be thought of as the amount of energy transferred per time in other words the amount of joules per second that are transferred and the name given to a Joule per second is a watt so you can solve for the power by finding the work divided by the time or the change in energy divided by the time and you can increase the amount of power by increasing the work done or decreasing amount of time it takes for that work to be done and just like energy and work power is not a vector so it's an example problem involving power look like let's say a box of mass M slid all the way down a frictionless ramp of height H and angle 2 theta as seen in this diagram and a separate mass M slides all the way down a frictionless ramp of height H and angle theta as seen in that diagram and we want to know how the average power developed by the force of gravity on the boxes compares for each incline so we use the formula for power powers the work done per time the work done on these boxes is going to equal the change in kinetic energy of these boxes which would equal the change in potential energy of the boxes but the mass of the boxes are the same the gravitational acceleration is the same and the height they fall from is the same so the work done on the boxes are equal but the time it takes for these boxes to slide down the ramp is not equal the mass on the steeper ramp will reach the bottom faster which means as a higher rate of power being done compared to the mass on the less steep ramp so even though the same amount of work is being done the rate at which that work is being done is greater for the steeper ramp compared to the more shallow ramp