https://youtu.be/fiiHmdjA450
PHYS 1101: Lecture Eleven, Part Eight
Now I want to move on to giving you my problem solving steps and walk you through an example. But let me start out and give you the big picture. So far we’ve considered two types of problems. For exam one we prepared ourselves for being able to solve kinematic problems that we say is just describing the motion. I’ve got time position velocity at an initial instant. Time position velocity at a final connected by acceleration. We have steps and we know how to solve those kinds of problems. Now we are moving in to a regime called dynamics where there are forces involved and we have to look at the implications of those forces on the acceleration and therefore the motion.
Depending on what kind of problem you have your solution strategy varies. We know if it’s only kinematics then our fundamental starting equations involve three sets of equations for the one-dimensional motion and we know what these equations are and we know what these problem solving steps are that helped us think through what these variables represented, what variables we had, what variable we had to focus to solve the problem. And these allowed us to solve a whole range of leaves, if you will, in my tree analogy.
Now in a problem if there is any mention of forces the right place to start, the base of your tree, the starting equation to use, the fundamental equations are Newton’s Second Law. The acceleration in that problem is a result of the forces divided by the mass. Meaning that the variable we are trying to solve for that represents the physical quantity that we are asked about in the problem, it will be represented in one of these basic starting equations.
So here’s where we start to solve any problem that has any mention of forces and what I want to give you now are the problem solving steps that are going to help you implement these equations to leverage them to solve your problem.
And of course it boils down to the same spirit we saw over here. I need to do some steps that slow me down and help me think about what the left side of this equation is the Ax and Ay component and what the right side of this equation is for this scenario for this particular leaf I am solving. That means what are all the horizontal forces? What are all the vertical forces?
So here are my steps. Any mention of forces and this is our starting equation. Fundamentally it’s this full resultant vector equation. I’ve got a tail to tip sum of vectors of forces. I divide that by the mass and that gets me vector acceleration. In order to do that exactly we need to work in the one-dimensional version of this equation. We need to work with the scalar components. That means I really have two fundamental equations to work with. The horizontal version, the horizontal components to this vector equations and the vertical version of this vector equation with the vertical scalar components.
Step one. Very common that people start mixing up objects when they work these problems and before you know it they’ve identified some forces that are on different objects. Focus on your one single object and you’re always asking what’s the acceleration of that object and what are the forces on that object. Don’t mix objects. When you use this equation must apply this to a single object.
Step two is having a slow down and think about what the left side should be of the equation. At this point you’re just trying to read the problem and determine if you have acceleration or not. If the left side is 0 the object is in what’s called equilibrium but you now know that a can be 0 if either the object is sitting still or moving along at a constant velocity.
Both of these are examples of equilibrium. You can’t tell from the acceleration being 0 you can’t tell from that alone which of these two scenarios that you have. A is either 0 or it’s not. If it’s not, you just want to read the problem and estimate with an arrow roughly in what direction is the acceleration because the direction of this vector will give you a picture of the direction that all of these forces have to add up to and point in. This is the same direction as f net.
And that’s step three is picturing the sum of the forces. Being sure that the sum is in consistent with the acceleration. Okay to get those forces draw yourself a free body diagram. Go through those steps I outlined for helping you to identify the forces because you have to get them all. If you miss one, if you miss a normal force, or if you miss a tension force or a friction force, then this equation is not true anymore and it won’t solve the problem for you.
Okay, I added step four here that check to see if there is any additional trigonometry that you can do because we do have to work with these two one-dimensional problems. The stuff aligned with the x axis as a separate equation from the stuff aligned with the y and so I made need for a given force, it can have a component in of x and y. So it may show up as the hypotenuse times the sine of the angle as one of these for example and the hypotenuse times the cosine of an angle. It might show up in that form in this equation.
Okay after you do some initial trig make yourself a list of your knowns and the variable that you want. And it might be helpful as we have done in the past to go over here and make a column for x and make a column for y. Put all of your x variables here and all of your y variables here and etc. I have this reminder; watch the signs because it is the scalar components that have to go in these equations. Sign means the direction and that’s important to keep track of, of course.
And then you are ready for step six. Work with those starting equations and try to focus in on the variable you need to solve for. And as always it’s good to have a last step here if you have time to go back and see if the answer makes sense to you. Check your units. That can flag any algebra mistakes you made in step six. Is the size reasonable? I remind you that one newton is roughly the weight. It’s roughly the m times g of a small apple, a Panera bread apple, when you order an apple as aside. It’s roughly a quarter; it’s actually a little less than a quarter of a pound.
So following these steps in particular the steps two or three. Identifying the acceleration in the problem and identifying the forces can be tricky so I have some quiz questions here for you to just get you to practice that and concentrate on that. The first few questions were just focused on what’s the acceleration as you read the problem. Your options are going to be it’s either 0, the object is in equilibrium, or pick one of these red arrows to represent the direction of the acceleration for this problem
Question 16 has to do with a heavy crate. Question 17 is a boy pushing a box. Question 18 focuses on the exercising of tossing a rock up in the air but let me emphasize that the scope of the problem is while the rock is still in your hand and your hand is moving up. You haven’t let go of the rock yet.
For all of these 16, 17 and 18, try to picture the motion. Try to draw yourself motion diagram. If your points are equally spaced, that’s constant velocity, so a is 0. If there is no motion at all, a is 0, but if your motion diagram shows this object speeding up or slowing down, then you know you have acceleration. You’ve got to pick one of these arrows.
Which one to pick? Remember if it’s slowing down, a has to point opposite to v. If it’s speeding up, it has to point the same direction.
The next collection of quiz questions has to do with identifying all the forces. This is another sticking point. Go through that and be sure you understand the identifying forces description that I gave the summary that’s earlier in this lecture.
For each of these scenarios I want you to identify the number of forces that are on the object. And let me point out that when you got a scenario as for example like this hand pushing down on this ball. Don’t get confused or worry too much about the details of all parts of that hand putting different forces on the different parts of the ball. Just represent for a given external object like this that’s in contact with the main object that you’re focused on. What’s the overall effect of all these forces? One force draw one blue arrow for this interaction.
And then another piece of advice what often helps me being sure that I have all my forces identified is thinking about what the visual balance is of the forces that I’ve identified and if it matches the acceleration. That can help you often realize you’ve forgotten a force. So as you do this think about the acceleration for each scenario and be sure that the number forces you have matches or it can be made. It’s reasonable that could match the acceleration that you have.
Let me say for Question 21 just as an example of including one force to represent an overall effect. Twenty-one is about a plane that’s flying and you’re trying to figure out how maybe forces are on the plane and it’s flying at a constant velocity and it’s fighting a strong headwind.
So let me help you out a little bit on this problem by just pointing out a few things. I want to illustrate how you want to represent an overall effect of one type of force and then also this visual check. Do my forces match acceleration?
It’s flying along at constant velocity so I know that the acceleration has to be 0. I’m going to jot that down. Where? Here. A is 0. Whatever forces I draw on this they have to balance. Okay, it’s flying. It’s near the surface of the earth. I know without thinking about anything else. Right away I have a force of m times g. The mass represented by the total number of molecules making this plane times 9.8. The length of this is fixed.
Okay. Drawing your circle around it. It first isn’t obvious that anything else is touching it or that I have to include any other forces but let’s think it through. If this were my only force this plane would drop like a rock. This force accelerating it downward would give me an acceleration down and it’s not. It’s flying along at a constant velocity.
So right away I know there’s got to be a force upward that balances the force of gravity. This upward force is the overall effect of the lift that comes from the airflow over the wings of this plane. It provides an upward force that keeps the thing up. It balances the forces due to gravity.
Now it tells you here that it’s fighting a strong headwind. That’s trying to indicate to you that you have some contact from the front that can’t be ignored. That’s the bombardment of these air molecules hitting the windshield of the plane the front edge of the wings etc. You have to put force into account for that. If you do that, you’re going to realize there is another force you have to include in order to make this statement true.
For this lawn mower question I want to emphasize that this isn’t one of these automatic powered lawn mowers that has self-propelling wheels. It’s just a manual push lawn mower. It’s cutting thick grass and that thick grass information is there to tell you that there is going to be that analogue of frictional force opposing the motion.
Okay, here’s two more. These should be fairly quick. Thinking first with the focus being on the cup for the object and then consider the saucer as being the object and then the last one I want you to do is one of these rocks on this sloped hill. Focus on just this rock. Ignore any possible static friction forces and tell me how many forces are on that yellow rock. In preparation for doing future problems think about the direction for each of those forces too.
And quiz Question 27, you are really putting a few things together to answer this. You have an elevator going straight up at a steady speed and you need to choose which one of these statements is true. These variables t and w represent the magnitudes of these forces. The tension force which is going to be the cable holding the elevator up and the force due to gravity on the elevator. So which of these statements is true?