A uniform soda can of mass 0.140 kg is 12.0 cm tall and filled with 0.354 kg of soda (Fig. 9-41

A uniform soda can of mass 0.140 kg is 12.0 cm tall and...

Key Concepts

Center of Mass

The center of mass is the point at which an object's mass is considered to be concentrated for the purposes of analysis. It is calculated as a weighted average of the positions of individual mass elements and is crucial for understanding the behavior of a system under gravity, as well as for analyzing stability and motion.

Composite Systems

A composite system consists of several parts or bodies that may have different masses and distributions. When computing the overall center of mass of such a system, each component’s center of mass is weighed by its mass, allowing one to treat the composite system as a single equivalent object for analysis of equilibrium and dynamics.

Weighted Average Method

The weighted average method is used to determine the center of mass of a system by multiplying each component’s position by its mass, summing these products, and dividing by the total mass of the system. This technique simplifies the analysis of complex bodies by reducing them to a single point that represents the overall mass distribution.

Effects of Mass Redistribution

When mass is added or removed from a system, the center of mass shifts according to the change in the mass distribution. Understanding this effect is important when analyzing systems where mass is being dynamically redistributed, as even a small change in mass locations or amounts can significantly affect the system’s stability and dynamic response.

Transcript

00:02 Our question says that we have a uniform soda can, and it tells us that the mass of the can, which i write as m sub c, is 0 .14 kilograms.

00:11 And it also says the height of the can is tall.

00:14 So i say that h is equal to 12 centimeters.

00:17 And it's filled in the 4 kilograms of soda, which i write that mass of the soda as m sub s.

00:24 So that's the mass of the soda when the can is full.

00:26 It says that small holes are drilled in the top and the bottom and that the loss of the metal from these holes is negligible.

00:34 And they're there to drain the soda.

00:36 It wants to know what the height of the center of mass of the can and the contents are for a, initially, before the holes are drilled, b, after the can loses all the soda, and then it wants to know what happens to the height as the soda drains, or excuse me, it wants to know what happens to h as the soda drains out.

00:56 So it wants to know what happens to the center of mass of h of the height of the can as the soda drains out.

01:03 And then for part d, it says that if x is the height of the remaining soda at any given instant, find x when the center of mass reaches its lowest point.

01:12 Okay, so starting out for part a, well, for part a, it's pretty simple.

01:19 We have the center of mass of the can.

01:22 We're assuming that the soda is full in the can, so we'll distinguish this as part a.

01:26 We'll call this h center of mass, com, because that's the abbreviation they use, is equal to the mass of the can times half the height, right, h over 2, because it says the can is uniform.

01:43 And if the can is uniform, then the center of mass of the can is just going to be half of it, right, just based upon the symmetry, right? and then also the center of mass of the soda, plus the mass of the soda, is also going to be h over 2 because the soda completely fills the can.

02:00 So h over 2, not h over s.

02:03 Because once again, you can just use the symmetry of the can to know that it's going to be half the height of the can, the center of mass.

02:09 And this is all divided by the total mass of the system that we're considering.

02:12 So this is going to be m subc plus m sub s.

02:18 Okay.

02:20 Well, we can pull the h over two out front in the numerator.

02:26 And this is going to be multiplied by m subc plus m sub s divided by m subc plus m sub s.

02:38 Well, clearly m sub c plus m sub s divided by m sub c plus m sub s is just equal to one.

02:44 So the center of mass of the can when it's full is just going to be h over 2.

02:48 H is 12 centimeters.

02:50 So this is just going to be six centimeters.

02:54 Okay.

02:55 And that's our solution to part a.

02:56 Part b wants us to do the exact same thing, but it wants us to do it when all of the contents of the can are completely drained out.

03:05 But if all the contents of the can are completely drained out, that is going to be in the exact same position as when the can is completely full because it's filled uniformly.

03:15 So that's pretty simple for part b, the center of mass of the can is still also just going to be com for center of mass as the abbreviation they use.

03:31 We'll stick with that abbreviation is once again just going to be h over 2, right? because if you go back to the previous part, part a, in this case, the mass of the can, m sub s, is just going to be zero.

03:46 So you're going to end up with m over m sub c.

03:49 And that's just going to be one.

03:51 So go back to where we're working here.

03:54 It's going to be h over 2, which again is just six centimeters.

04:05 Now, part c wants to know what's going to happen to the center of mass as the soda drains out.

04:13 So while it's in the process of draining, what happens to.

04:16 To the center of mass.

04:17 So we're just going to write a little description of what's going to happen here.

04:20 I think that's the best way to do it.

04:22 So as the soda drains out, the center of mass will decrease towards the bottom of the can.

04:38 Now, the reason this is true is because initially the center of mass, which is part a when the can is full, is at h over 2 or 6 centimeters.

04:45 But the soda weighs more than the can.

04:47 So as the soda starts to drain out or more of the soda starts to fill the bottom of the can, more and more of the center of mass is going to be pushed down towards where the soda is draining because more and more mass is at the bottom of the can since that's where the soda still is.

05:02 But eventually we know from part b that once it's all drained, the center of mass is going to be back at the top.

05:09 So as the soda is draining out, the center of mass will decrease towards the bottom of the can until, oh, i ran out of space here, so we'll continue on the next line.

05:23 Bottom of the can until a low point for the center of mass is reached.

05:37 It will then start to rise, start a new line, then we'll start to rise until h over 2 is reached, or six centimeters, right? so it's going to start to, the center of mass is going to lower as the can drains towards the bottom until it reaches some arbitrary low point, which you could figure out.

06:08 And then it's going to start to increase again until it reaches six centimeters.

06:12 So that's the behavior, right? and then for part d, it wants to know if x is the height of the remaining soda at any given instant, find when x or find x when the center of match reach, mass reaches its lowest point.

06:25 And that's the lowest point that i was discussing here in part c.

06:27 So we'll open up for part d.

06:31 Okay.

06:33 So we're going to call this new mass.

06:36 Or we're going to call the mass when it reaches the low point m subprime or in s subprime.

06:41 So this is the mass of the soda when it's reached its lowest point.

06:45 And that's equal to the mass of the soda times x over h, right? where h is the mass of the can.

06:54 So when x, which is the height of the soda, is equal to h, the soda can is full.

06:59 And we have the full mass of the soda.

07:01 We just get back to m sub s, which is what we expect.

07:04 Also, when x is equal to zero, that means all the soda's been.

07:07 Drained out of the can, and so ms of s prime is zero, which is also what we expect.

07:11 So this equation works.

07:12 This is a working equation here.

07:14 So now the center of mass h -c -m or c -o -m, sticking with their notation, is equal to the mass of the can times h over 2, which is just the center mass point for the can with no soda in it, plus ms -prime...

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