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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) According to a simplified model of a mammalian heart, at each pulse approximately 20 $g$ of blood is accelerated from 0.25 m/s to 0.35 m/s during a period of 0.10 s. What is the magnitude of the force exerted by the heart muscle?
(I) What force is needed to accelerate a sled (mass = 55 kg) at 1.4 m/s$^2$ on horizontal frictionless ice?
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welcome to our next unit on alternating current in this unit. Instead of considering the classic circuits that we have, where we apply some e m f over some load and we ask what happens, we're going to change it so that we now have a source which will be denoted like this, applied to some load. And what this source says that we have an IMF that varies as a function of time. Here, we'll call it, eh? Not as our amplitude multiplied by Kassian of omega t plus five. Okay, Not every book will write out this function this particular way. Some books will use sign. Some will ignore the phase constant. I prefer to include the phase constant and to use cosine. Um, the reason being that Kassian often is where we want to start with the voltage and the fi allows us to account for that. If we don't actually start at a maximum applied E m f s. So you can choose which everyone works. The physics and the mathematics will all work out to be exactly the same. Now the reason we're interested in alternating current is because it's much more riel world example remember that when we discussed generators, we had a loop inside of a permanent magnetic field and we caused this loop to rotate around and around. And we did this by spinning a turbine that it was attached to either with wind or with water or with steam. So when we spun this around and around, what we found was that the induced current in this ring and this loop looked on awful lot like an cosine of Omega T plus five sort of function. So this is actually going to be more along the lines of what we are literally producing at our power station. It's an alternating current, and then it is transmitted to our homes as an alternating current source. So this is much more what we see in our homes. Now, some devices in your home will have DC converters that is direct current converters so they'll work more like this. But generally speaking, ah, lot of our appliances devices will work on a C. So understanding how these work and how they interact with the different circuit elements that we've learned so far, uh, can be really, really helpful for us. Another thing we need to talk about before moving on is how we're going to visually represent this. Now you've seen many times before the plots ease function of T against tea and it'll look something like this. Remember that important quantities air here are going to be the amplitude which in our equation, eyes e not finding the period, which is the time from here to here, because we know that period is equal to two pi over omega and we can also keep a nigh on the phase constant. For example, if we saw the functions start here instead, we know that it had shifted by pi over two. So we would have to write cosign of omega t plus pi over two because it's been shifted to the left Okay, s so we can pull all these things out of here and that's very helpful. But very commonly, instead of using that sort of coordinate system, we will use what's called a phaser to draw out are alternating current source or current eso are alternating voltage or our alternating current. What we'll do is then with a phaser is we have a vector which, when it is at the zero angle is a length according to your amplitude e nut, and then some time later it will move up by an angle. Omega T plus. If I noticed, if I is not zero, then we wouldn't start here. But the idea, then, is that this vector maintains a constant amplitude as it goes around at different angles. And so the angle can always be found as a functional mega T plus fi measured from the X axis, the positive X axis and it goes around and around in the counter clockwise direction. Uh, now if we want to find the actual applied voltage, replied e m f at any particular point, What we need is the horizontal projection onto the X axis. So in this case, that would be e not times cosine of Omega T plus five. You can see it's the exact same equation we had before. This is why some people prefer to use the phaser to represent they're alternating current or alternating voltage