section 7.3 work done by the gravitational force so we can calculate the work done by specifically gravitational force if we just take our work equation and plug in our gravitational force right so instead of the f we're going to plug in mg because that's what our force is here times the displacement times cosine of the angle between the force and the displacement so this is really just looking at a specific subset of the previous equation not all that new but we can check a few common cases right what about when we have a rising object so the object is going up into the air well in that case the displacement is going to be positive what direction is the gravitational force acting on gravitational force is pulling down on it right with the mg force so what's the angle between the displacement and the force it's 180 degrees and so we get negative one for that cosine and that tells us that the work done by gravity is negative mgd for a rising object if instead the object is falling with some velocity like an apple or a ball falling through the air well notice that is now in the same direction as the force of gravity mgd or mg so the angle between displacement and the force is zero and we have cosine of zero is positive one and we can calculate the work done by gravity is now positive mgd what does this mean well think about if an object is rising in the air then it is losing kinetic energy right so the work done by gravity is its removing energy transferring it from the object as the object starts to fall it moves faster and faster it gains kinetic energy and so now the work done by gravity is actually transferring energy to the object and that's why the work done is positive versus negative so the signs here really do matter all right now we could also think for a moment not just of tossing a ball but what if we're lifting or lowering an object right well either way you can imagine we have like we're weightlifting here throughout the motion we are going to be supporting that bar right keeping it from falling and crushing us so even when we're raising it and when we're lowering it we're applying an upward force to the object so that would be see if i can get a sketch going up here so if we have say an object the force of gravity is coming down mg and we're applying a force to support it to fight gravity so that'd be f sub a for the applied force of what we are bringing to the table so we know that the change in kinetic energy okay final minus k initial is equal to the net work done if we have two forces that are acting in the direction of the motion then the net work will be the work done by the applied force plus the work done by gravity so this is a handy subapplication again of our overall wart kinetic energy theorem the change in kinetic energy is equal to the work that's the much better general equation but for this sub case we can look at the limiting case that what if the kinetic energies are zero so if we hold an object and it's not moving and then we lift it up and it at the top is not moving well then at the end as well as at the beginning the kinetic energy is zero so the net change in kinetic energy is zero as well the net work work applied plus work done by gravity is equal to zero so that tells us that the work applied is always going to be negative the work done by gravity because the force is acting in the opposite direction so it's going to have the opposite sign of work in other words for an applied lifting force you can find that the work applied is negative mgd cosine of phi i don't really like this equation so i don't highly recommend using it because it depends on this angle being between gravity and between the displacement not the applied force but i do think it's reasonable that the work applied is negative the work done by gravity that's a better equation to use it seems a little safer to me you can calculate the work done by gravity and know that the work done by the applied force is just going to have the opposite sign a useful thing though is that this equation applies regardless of path so i talked about a simple example of lifting straight up right but i could have also walked for a while with the bar i could have lowered it further and then lifted it up higher and it would end up being the same amount of work done by the applied force versus by gravity pretty wild and that's in part because of the specific definition of work that we have for physics all right so here's a few uh examples showing the applied force and the force of gravity right and so in every case whether the object is being moved upward or lowered so the upward displacement on the top and downward displacement on the bottom we see that if there the forces are in the same direction that force is doing positive work if the force is acting opposite the displacement that force is doing negative work and so which one's doing positive work will depend on if you're moving up or if you're moving down note that the works don't have to be equal you might be doing more work in the applied for in applying a force than the work done by gravity but in that case the change in kinetic energy will not be zero right so this is we'll only see that the works are equal and cancel if the initial and final kinetic energies are equal so works are unequal you'll need to know the difference between the initial and final kinetic energies in order to solve for the work so it might tell you that it's at rest to start and then it's going at two meters per second well that can tell you the change in kinetic energy and that will tell you the net work so there we have it let's look at a brief example here you're a passenger being lowered down in an elevator which forces do negative work which forces do positive work so i want you to think for a moment about this come up with an answer to both questions before you move on so go ahead and pause now all right well first off we need to think about the forces in the elevator right so one of our forces is going to be gravity can't get away from it it's always pulling us down so we have the force of gravity pulling us down what other forces are acting on you in the elevator other force is the elevator cable the tension that's supporting the entire elevator or if you don't want to think of the elevators the object of interest you could have talked about the normal force that's supporting you that would be equally valid depending on your perspective so i'll note that one too so the normal force would be if you're talking about the forces acting on you as a person passenger the tension would be if you're talking about the forces acting on the elevator now to figure out what forces are doing positive versus negative work we need to write down our displacement so it says we're being lowered down so that tells us our displacement is downward so the forces doing positive work are the ones that are in the same direction as that displacement that's going to be gravity is doing positive work which forces are doing negative work it's going to be the ones that are going against the displacement so either the normal force or the tension as shown so here it's focused on the elevator cab so it says tension does negative work the force of gravity does positive work in other words the force of gravity is helping you move down while the tension is supporting the elevator so that you do not move down too quickly which is a good thing we're very glad for elevator cables when you're in an elevator so we'll wrap up the section right there