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University of Michigan - Ann Arbor
University of Winnipeg
(I) A 7150-kg railroad car travels alone on a level frictionless track with a constant speed of 15.0 m/s. A 3350-kg load, initially at rest, is dropped onto the car. What will be the car's new speed?
(II) A person has a reasonable chance of surviving an automobile crash if the deceleration is no more than 30 $g$'s. Calculate the force on a 65-kg person accelerating at this rate.What distance is traveled if brought to rest at this rate from 95 km/h?
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welcome to our section on two dimensional dynamics. So in this unit will be looking at all different forms. We've been looking at one dimensional dynamics, and now we're going to start looking at two dimensional dynamics. Eso in the most basic sense. This means that we're going to be looking at forces that air in both the X and why directions and affecting the motion of an object of the position of an object as a function of time for both of those things. Um, the best example we've come across so far of this previously was projectile motion. We threw a ball only talked about how it would have a parabolic arc with one important assumption here. We just talked about drag forces, and this was in a zero air resistance environment. If we're not in a zero air resistance environment, then this is going to change, and we would want to work on and figure out how that would. That would function as in two dimensions, as opposed to just and thinking about zero air resistance. So we will briefly consider this problem because of the complicated native nature and the differential equations involved with solving this It's generally not covered heavily in introductory physics courses, and so we'll just spend one example talking about it. Um, now, other things that we're going to cover, though, is circular motion. Very important again. Remember, we had some equations that we had set up for that before, As equals, R theta and gentle velocities are times omega and tangential. Acceleration now is our times, Alfa. So we have position, velocity and acceleration, linear and angular here, and we're going to start to look at what causes this stuff to happen. We already have one idea we remember. We also had centripetal acceleration, which we found was equal to V squared over R or omega squared. Times are and we already know that F is equal to mass times acceleration or at least some of forces, is equal to mass times acceleration. So if we see a centripetal motion, that means that our F net here are some of force is going to add up to a centripetal force which will equal mass times a centripetal acceleration or, in other words, M B squared of R or M Omega squared are That's a lot of equal signs there, but these things are all equal to each other for the situation pictured here, where we have an object moving Onley in centripetal motion. On the other hand, we could cause non uniforms circular motion by causing some sort of acceleration, some tangential acceleration like this one here, some tangential acceleration which will cause an Alfa. An acceleration in Alfa would change omega, which would mean a different VT, which would change your centripetal acceleration. So these things are tied to each other, though not exactly the same. So what will say is that you might also have an S net that has a tangential force in it, which will be equal to mass times 10 gentle acceleration. Um, now we have another way. We'll look at this in a couple of units here, but for now, this is how we're going to think about it. We have centripetal forces that are associated with centripetal acceleration and tangential forces that are associating with tangential accelerations. And that's what how we're going to go about kind of trying to analyze the dynamics of these situations. We may change tangential velocity by giving it a tangential force, or we may change it by thinking about? We have this centripetal force. What if we change centripetal force? Well, that would also change V squared. What if we changed our well? Our is related to these things, so it changes as well. If FC changes in VT changes, then Omega changes of Omega changes it means there was an Alfa, which means there was a tangential acceleration and all these different things. So Thies too forces are really closely tied to each other. And we need to make sure that we keep in eye on mostly speaking, though, if it's moving at a constant omega, that means we have a centripetal force. But no tangential force. If we have a changing omega, means we have both a tangential force and a centripetal force because we still need to continue to go in with going a circle with this center pointing motion. So there definitely is a centripetal force here. Um, we're gonna analyze some of the classic physics problems and how to do this. Things like cars going around a turn or even a car going around a banked turn, like on a racetrack. Um and especially we're going to consider planets and their motion around the sun because this is also a circular motion. Mostly, obviously, it's not perfectly circular, but it's pretty close, and we can get a lot of information simply by analyzing this motion with two dimensional dynamics. Um, as we are about to dive into, um now with two dimensional dynamics, as you may have guessed, watching uh, in the previous screen that it's gonna be more advantageous to us to use the data. Omega and Alfa most of the time occasionally will throw in a centripetal acceleration, which would be related to a tangential velocity on beacon get S, N V T and A T from these. But generally speaking, these air the variables of choice. It is also possible that we could look at X and why they just become very, very complicated very, very quickly, because as you try to work with them, they become functions of each other because you have motion in both the why and the X that may depend on each other. So because they become dependent on each other than math becomes much more involved, end up with these parametric equations. So it's much simpler to focus on circular motion here. Our first video will consider an example of projectile motion with Y and X, but then we'll move on from there and start considering, uh, objects moving in a circular pattern.
Equilibrium and Elasticity