This is module ten of two dimensional dynamics. Today's learning outcomes are to calculate the work done by gravity. Last time we did the work done by a linear spring. And also calculate the work done by friction. And then finally solve engineering mechanics problems for particles, systems or particles using the work energy principle that we've developed. So work on by gravity, work on by gravity is force times distance. In this case, the force is the weight and the distance is the change in altitude of the mass center of the body and so let's take this golf ball here. You saw in my introduction video. I'm an avid golfer, so I had this golf ball lying around my my office. And so the work done is going to be mg times the change in altitude. And it's conservative, so it doesn't matter what path I take. Okay the same amount of work is going to be done regardless of the path and if I move around and I come back to the same location, the net work done is zero. And there's no change in energy, so it's a conservative, force. Okay. Work done by Coulomb friction is non-conservative. Here I have the weight of my body the normal force, the friction, Coulomb friction is related to the normal force by mu times N. It's always going to oppose motion, and so in this case, in going from position one to position two, I've got this friction force that, here's the coefficient of friction. And so let's go over and look at a demonstration here. Okay, if I'm move in this direction, friction at the bottom is going to oppose motion. Okay, it's going to be equal to mu times N. however, it's non-conservative because if I go back in the other direction, the friction switches direction and I continue to lose energy. So, when I get back to the original position, the net work is not equal to zero. Energy has been lost. It's non-conservative. And that's work done by friction. And so we have work equals minus the force times the distance, in this case we say it's gone through a distance, d, the friction force is Coulomb friction mu times N. And so work due to friction is always negative because it always hinders the motion, it always opposes the direction of motion. So let's go ahead and do a, a, a problem here. Let's find the maximum force in the spring for this situation. I've got a block it's sliding down the plane, the coefficient of friction is 0.3, it's got an initial velocity of 30 inches per second and I've put g on here, the acceleration due to gravity. Since I'm working in inches and seconds, I've changed 32.2 feet per second squared, the acceleration of gravity to 386 inches per second squared to be consistent with units. And then it hits this spring and the spring has a spring modulus of a 100 pounds per inch. And as it compresses, we want to find the maximum force in the spring, and so we're going to use the work energy principle. Work done by external force is change in kinetic energy. And so, as far as the kinetic energy is concerned, first of all I'd like you to write down what is vC t1 and what is vC t2 for this situation. And, actually let's talk about the magnitudes since we're not working with vectors, we're going to talk about the magnitude of the velocity. And come on back once you have that done. So. We were given, vC t1, it's magnitude is 30 inches per second. That's the speed. And as far as vC 2, if I want to find the maximum force in the spring, that's when the block gets down and it gets to zero velocity. That's the maximum that the spring is going to be compressed and that'll give us the max force. So, vC t2 will be zero. So now we've taken care of the right hand side, the kinetic energy. Now draw a free body diagram of the block so that we may calculate the work done by the various forces acting on the block. And after you've done that, come on back. Okay, here's the free body diagram. I've got my weight force, I've got my normal force, I've got my friction force, and I've got my spring force. So let's start by looking at the work done by the spring. Remember, it's going to oppose motion, so it's negative minus k over 2 is equal to the change in, the deflection of the spring. In this case, delta t1 is 0, we're going to say it starts out in the unstretched position and when it gets down to where the spring is maximally compressed, we'll call that distance delta. So what I have here is. Work of the spring is minus 50, excuse me 100, over 2, times delta squared or minus 50 squared, delta squared. Minus k over 2, delta squared of t2 is Delta, this Delta. Capital Delta minus delta t1 is zero. All right, let's do he same thing for the work, due to gravity. I want you to go ahead and write that down. Come on back and we'll do it together. So, we know what the, we know what the weight is. It's, lets see it's given to us as 25 pounds. We need to know the change in altitude. So let's draw a picture here. We know that this block is going to slide ten inches plus however much it compresses. So 10 plus delta. The change in altitude in going from the initial position to the final position will be r, and so I can use similar triangles. Here's the 3, 4 triangle so I've got the three, four, five side. That means that this r is to 10 plus Delta as 3 is to 5. And so the change in r here is going to be 3 5ths. Whoops. 3 5ths 10 plus Delta. Okay. And, the last thing I want to do is the work of friction. Minus f times d. And so that's going to equal. Do it on your own, then come on back. That should be minus f, which is 6 pounds, times the distance that it slides, which we said was 10, plus Delta. So now I have all of the work terms. I've written them up here. Work to the spring is, we said was minus 50 delta squared. There is work done to gravity, work done by friction. We know that the final velocity was zero. The initial velocity was 30 inches per second squared. I can substitute that into my work energy principles. So, total work is on the left side, change in kinetic energy is on the right hand side. That becomes a quadratic in delta. So, I find out that delta is, this value. Physically it only ma, ma, makes sense to have a positive value for the co, compression of the spring. When it's maximally compressed, the force in the spring will be k times this delta which is 1.636 inches. We were given that k was 100 pounds per inch, so we see that the more, max force in the spring, is 163.6 pounds. And we've done a work energy principle problem. And so I'll see you next time.
