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(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?
(I) What is the weight of a 68-kg astronaut ($a$) on Earth, ($b$) on the Moon ($g =1.7 m/s^2$) ($c$) on Mars ($g = 3.7 \,m/s^2$) ($d$) in outer space traveling with constant velocity?
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Welcome to our next section on non uniform, non uniform. Circular motion. Okay, Non uniform. Circular motion. So, as opposed to previous conversations where we've only talked about Seda in Omega. Now we're going to add in our angular acceleration Variable Alfa. So again, remember, this is kind of equated with linear acceleration, linear velocity and linear position. All of these are vectors. I explained the right hand rule last time for Omega. It works the same for theta and for Alfa as well, where we will be treating our circular motion variables as one dimensional variables. Now, um, it turns out that when we write thes we found relationships that we had found before we had omega equals Delta theta over Delta T, which allowed us to get that data as a function of time is equal. Thio they did not plus omega times time. Well, it turns out that there is a similar relationship between angular acceleration and angular velocity is their abuse, as there is between linear acceleration and linear velocity. So Alfa is equal to a change in omega over time, which we can also find that omega, as a function of time, is equal to Omega, not plus Alfa Times time. We also find our other Kitimat equations. Principally, that data can be written as they do not plus omega, not times time plus one half alfa t squared and also that omega squared is equal to omega, not squared, plus two Alfa Delta theta. So they should all look familiar to you because they were previously written as linear equations. Remember, we had X as a function of time is equal to X not plus bt. Remember these equations all require a constant velocity. And then we had these. A function of time is equal to the knot plus eight times to you Remember, these equations all require a constant acceleration. Then we had X is a function of time is equal to x, not plus be not t plus one half A t squared and B squared equals be not squared plus to a Delta X. So what we're doing then is we're really just doing Kinnah Matics, except we've considered a different access. We're considering rotational axis as defined by the right hand rule and we'll be able to use our same strategies that we used before to solve for linear Kinnah Matics in order to solve for thes rotational Kinnah Matics or angular Kinnah Matics. Beyond that, there's not a whole lot of difference. Remember, we have the right hand rule defining direction. Generally speaking, I'll pick out of the screen to be considered as positive direction and into the screen, to be considered the negative direction. It's difficult to draw this on a two dimensional surface, but if I were to rotate it and say I have all right hand rule that says I'm rotating this way, then I would say that this is my positive direction and this is my negative direction. In other words, you can also say that counterclockwise is positive and clockwise is negative Now moving on. Then we can think about all of these in the same way we did before. Except we only have one dimension, which means we don't even have to worry as much about having X and y in all of our equations. Separately, we only have to worry about thes, which can make it a lot simpler, but we still have the same constraints. As I said before. These two equations assume a constant omega, and these equations assume a constant Alfa as before. If we have an Alfa that changes thing, we can consider constant alphas in different units of time. For example, if we have a fan that we turn on and it starts rotating and accelerates up to some rate, and then it stops accelerating, which reaches its high, its velocity, then we can turn it off and it will accelerate back down. In that case, we'll have. Omega is a function of time will look something like this. It will start a zero. We turn it on, it starts to increase, it hits its peak and it sits there. Then we turn it off and it goes back down. We definitely wanna be thinking graphically about how these variables they're going to look and we'll be able to use our tools of cinematics and then also integration and derivatives in order to solve For how maney rotations did the fan go to what was its acceleration Any particular time? We remember that just like before we have omega is equal to the time derivative of theta and Alfa is equal to the time derivative of omega, which means that data function of time is also equal to the integral of Omega with respect to time and Omega is equal to the integral of Alfa. So we have the same relationships we had before. We're just supplying them in a slightly different context, so let's move on to the examples.
Newton's Laws of Motion
Applying Newton's Laws