Good morning. Today we are going to review the work, energy, and power portions of the AP Physics C Mechanics curriculum. ♫ Flipping Physics ♫ Bobby, what is the equation for the work done by a constant force? - Work done by a constant force equals the dot product of the force doing the work and the displacement of the object. - Or force times displacement times the cosine of the angle between the force and the displacement using only the magnitudes of the force and displacement vectors. - That's what the dot product means. - Yeah. - Work is a scalar. - Let's do a simple example. If the force acting on an object is 2.7i minus 3.1j newtons and the displacement of the object is 4.6i meters, Billy, what is the work done by the force on this object? - Work equals the dot product of force and displacement, so it equals the dot product of 2.7i minus 3.1j, and 4.6i plus 0j. Multiply the i's together, multiply the j's together and take the sum. So it is 2.7 times 4.6 plus the negative 3.1 times zero, or 12.42, which is twelve with two significant digits. Oh, and it's in joules. - Notice the dot product multiplies the component of the force which is in the direction of the displacement of the object, with the displacement of the object. In this particular case, the component in the direction of the displacement of the object is 2.7 newtons multiplied by the displacement of the object of 4.6 meters. The component normal to the direction of the displacement in this particular case, 3.1 newtons in the negative y direction, does no work on the object, again because it is at a 90 degree angle to the displacement of the object. And yes, Billy, the units for work are joules. Bobby, what is a joule? - A joule is a newton times a meter. - And a newton is a kilogram meter per second. - Squared. - A newton is a kilogram meter per second squared. - The work done by a force which is not constant uses a different equation. That work is equal to the integral from position initial to position final of the force with respect to position. This is called a definite integral. Which simply means the integral has limits. In this particular case, from the initial position to the final position. Bo, the derivative represents the slope of a function. What does the integral, or what we also call the antiderivative, represent? - An integral, or antiderivative, represents the area under the curve, and the area under the curve means the area between the curve and the horizontal axis. An area above the horizontal axis is positive. The area below the horizontal axis is negative. - Notice how we have two different equations for work. We use one equation for work when the force is constant. We use a different equation for work when the force is not constant, and that equation uses an integral. This is going to happen a lot in this class, where we use one equation when the item does not vary. We use a different equation for that same item when it does vary and that equation uses an integral. Please pay attention to that. Now let's talk about the force caused by a spring: Hooke's Law. The force of a spring equals negative kx. Bobby, can you please clarify all of these letters? - K is the spring constant, and it is a measure of how much force it takes to compress or expand the spring per meter. Delta x is the displacement of the spring from equilibrium position, or rest position, which is where the spring would be if it were not being compressed or elongated. The negative has to do with the direction of the force of the spring. It means the force of the spring is opposite the direction of the displacement of the spring, and the units for the spring constant are usually newton meters. - Actually, it's newtons per meter. - The spring constant is in newtons per meter. Joules are newtons times meters. Thanks. I always mess that up. - You are welcome. - Billy, which of the two work equations do we use when determining the work done by the force of a spring? - Well, that would depend on whether the force caused by a spring is constant or varies. - The force caused by a spring changes with position, so it is not constant. - Right, therefore the work done by a spring equals the intergral from position initial to position final of the force of the spring with respect to position. - Bobby, please determine the work done by a spring. - Well, the equation for the spring force is the negative of the spring constant times the displacement from equilibrium position. The integral of x to the first power with the respect to x is x squared over two, so we have negative kx squared over two from position initial to position final. Substituting in our limits gives us way too many negatives, so I'm going to factor out a negative one, and one half kx squared is elastic potential energy, so that is the negative of elastic potential energy final minus elastic potential energy initial, or the negative of the change in elastic potential energy of the spring. The work done by a spring force equals the negative of the change in elastic potential energy of the spring. That's pretty cool. - Previously we derived to the net work kinetic energy theorem. We're not going to do that derivation right now, however you're more than welcome to review it by clicking on the link which just appeared. - [Mr. P.] It was during that derivation where we defined kinetic energy. Kinetic energy equals one half times the mass of the object times the velocity of the object squared. The AP equation sheet just uses the capital letter K. I prefer capital KE so that you don't confuse the spring constant with kinetic energy. Gravitational potential energy equals the mass of the object times the acceleration due to gravity times h, the vertical height above the horizontal zero line. Please, don't ask me why the symbol for gravitational potential energy is a capital U. I don't know. - [Mr. P.] Please remember whenever you use gravitational potential energy, you have to identify the horizontal zero line. This is the reference line for the vertical height h. The equation on the AP equation sheet is instead in terms of the change in gravitational potential energy. Energy can neither be created nor can it be destroyed. Therefore, in a system which is not isolated. The change in energy of a system equals the sum of the energy which is transferred into or out of the system. Bo, this is important. Could you please review it for us? - When we have a non-isolated system, which means a system which is receiving energy or having energy taken away from it, the change in energy of the system equals the addition of all the energy which is transferred into or out of the system. It actually makes a lot of sense if you think about it because you cannot create or destroy energy. The energy has to come from or go somewhere. - Right. Now, if the system is isolated then no energy is transferred into or out of the system, therefore the change in energy of the system must be zero. The change in energy of the system equals the change in mechanical energy of the system plus the change in internal energy of the system which add up to zero. The change in internal energy of the system equals the negative of the work done by non conservative forces. And remember, the work done by a conservative force does not depend on the path taken by the object. The work done by a nonconservative force does depend on the path taken by the object. Therefore, the work done by nonconservative forces equals the change in mechanical energy of the system. Because the only non conservative force that I know of is friction, I usually write this equation as: The work due to friction equals the change in mechanical energy of the system. And remind me, Bobby, what needs to be true for the work due to friction to be equal to the change in mechanical energy of the system? - There needs to be no energy transferred into or out of the system. - The system needs to be isolated. - Don't confuse the work due to friction equals change in mechanical energy equation with the net work equals change in kinetic energy equation. - They do look similar. - Net work equals change in mechanical energy is not the same as work due to non conservative forces equals change in mechanical energy. - And if the system is isolated and no work is done by friction, then the change in mechanical energy equals zero, which works out to be conservation of mechanical energy. Class, remind me, what do you need to identify whenever you use conservation of mechanical energy or work due to friction equals change in mechanical energy? - Initial point. - Final point. - Horizontal zero line. - Remember, just like work, all of these forms of mechanical energy are in joules. Now let's talk about power. Power is the rate at which work is done. Therefore, average power equals work over change in time. Instantaneous power equals the derivative of work with respect to time. I will point out that the equation for power in terms of work on the AP equation sheet is: Power equals the derivative of energy with respect to time. Now, substituting in the equation for work done by a constant force gives us the derivative with respect to time of the dot product of the force and the displacement of the object. But the force is constant, so this is really just the derivative of the position with respect to time, which is velocity. Therefore, the power delivered by a constant force on an object in terms of velocity is the dot product of force and velocity. Bo, what are the SI units for power? - Power... Is in horsepower? - Horsepower is the English unit for power. Watts are the SI unit for power. - Right, and 746 watts equals one horsepower. - Watts are joules per second. But we don't have to memorize that there are 746 watts in one horsepower, right? - You are correct that you do not need to remember how many watts are in a horsepower. What you do need to remember is that every derivative is an integral, or an antiderivative. For example, power equals the derivative of work with respect to time can be rearranged to dw equals power times dt. Taking the definite integral of both sides gives the change in work equals the integral from time initial to time final of power with respect to time. Which means if you have a graph of the power exerted on an object with respect to time, the area under that curve will be equal to the change in work exerted on that object. Which could be very helpful. So please remember, every derivative can be rearranged as an integral, and every integral can be rearranged as a derivative. Billy, what is the equation which is not on your AP equation sheet which has to do with conservative forces? - There's an equation which has to do with conservative forces which is not on our equation sheet? - I did not know that. - It's a pickle. - Pickle? A conservative force equals the negative of the derivative of the potential energy associated with that conservative force. As a side note, you have to know this equation because much of the time, when the phrase "conservative force" is used on the AP Exam, this is the equation you need to use. Capeesh? - Capiscilo! - Yeah. - Ich verstehe. - For example, for a spring, the force of a spring equals the negative of the derivative of the elastic potential energy associated with that spring with respect to position. We can substitute in the equation for elastic potential energy, and Bo, could you please take that derivative? - The derivative of x squared with respect to x is two times x. 1/2 times two is one. So the force of a spring equals negative k times x. But we already know that, don't we? - [Mr. P.] Well Bo, that is my point. - Right. - Billy, could you please do the same thing with the force of gravity? - The force of gravity equals the negative of the derivative of gravitational potential energy with respect to position. However that position is going to be in the y direction because the force of gravity is in the y direction. For gravitational potential energy, we can substitute in mass times acceleration due to gravity times vertical height above the horizontal zero line. Although, let's use y for that instead of an H to match the variable we used for position. Mass and acceleration due to gravity are constants, and the derivative of y with respect to y is just one. So the force of gravity equals the negative of mass times acceleration due to gravity. Why is it negative? - The force of gravity is always down. That's why it is negative. - Oh, okay. That makes sense. - Now let's talk about neutral, stable, and unstable equilibrium. This ball is in neutral equilibrium because the gravitational potential energy of the ball remains constant, regardless of its position. So the graph of the gravitational potential energy of the ball with respect to position is a horizontal line because the gravitational potential energy is constant. This water bottle is in stable equilibrium because its gravitational potential energy increases as its position moves away from the equilibrium position. This is because the center of mass of the water bottle goes up as the position goes away from equilibrium. In other words, the water bottle naturally returns back to the equilibrium position when it loses gravitational potential energy. This dry erase marker is in unstable equilibrium because its gravitational potential energy decreases as its position moves away from the equilibrium position. This is because the center of mass of the dry erase marker goes down as the position goes away from equilibrium. In other words, the dry erase marker naturally moves away from the equilibrium position when it loses gravitational potential energy. That completes my review of work energy and power. Next, feel free to enjoy my review of integrals in kinematics for AP Physics C. Or you can visit my AP Physics C Review webpage. Thank you very much for learning with me today. I enjoyed learning with you.