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A metal rod of cross sectional area $1 \cdot 0 \mathrm{~cm}^{2}$ is being heated at one end. At one time, the temperature gradient is $5-0^{\circ} \mathrm{C} \mathrm{cm}^{-1}$ at cross section $A$ and is $2 \cdot 5^{\circ} \mathrm{Ccm}^{-1}$ at cross section $B$. Calculate the rate at which the temperature is increasing in the part $A B$ of the rod. The heat capacity of the part $A B=0-40 \mathrm{~J}^{\circ} \mathrm{C}^{-1}$, thermal conductivity of the material of the $\operatorname{rod}=200 \mathrm{~W} \mathrm{~m}^{-1}{ }^{-1} \mathrm{C}^{-1}$. Neglect any loss of heat to the atmosphere.

In this question we will learn about some basic concept of the heat transfer. The question is we have the length of the load. That is one m. The girl sectional area is 100 cm square. That we can convert it into Middle Square right? And the one hand is 100 C. And the other than this uh Jody descend together so we have to find the quantity of he transmitted to the door. But you may not. The thermal conductivity has given us 100 watt per kg by Calvin. So we can recall that cuba I. D. That is the letter of transfer. This Okay data and minus data. Do divided by L. So thermal conductivity is 100. The girl sectional area is 100 into 10 for minus fault. The two temperatures. The of the two ends would be 100 0 degree. Right? And the land after all this one m. So when we saw this we get cuba ds 100 jule per second. But we have to convert it into minutes. Jewel permanent. Right? So 100 multiplied by 60. That would be equal to 16 to 10 power three jules per minute. Mhm. Like so this is the date of the transfer. The angel per minute Hands. After the given four options taken adoption would be BB

All right. So this brushing me are after a both the area on a curse. Um, so in question, they were asked with the area under the curve between time with zero and time in calls for five. So I mean, we consider this to be your time, because five, uh, and we can kind of put this into three areas. So we got two rectangles. They're trying. So the first rec time goal is to buy two. So the area is gonna be, um, four plus on mrs. To this, rectangles will be to buy three go five, uh, plus six. Plus. Now, this triangle the the height. Ah. Is there three and the, um yeah, I just three in length is three. So, um, the area is gonna be 1/2 times nine, which is 4.5. The area is 14.5. So in part B were asked about the area between zero or sore between five and 10. So between five and 10 again are basically going to do the same thing. We got a little triangle, appear we've got a nice rectangle and another nice rectangle. So the smaller rectangle it is won by five. So area, it's areas five The triangle It's one by one. So it's area his 0.5 by one by one. I mean base and I have one. And then the big rectangle it is four Ah, by six. So the area is 24 so total area is 20 9.5. So now we and by so yes, O c So total from 0 to 10 which just add the 055 10 up. So that is gonna give us 44 and then lastly, in part D that ask us about the center of mass. So, um, the what this is really asking us is, um what, uh, X value is, um, half. Um, I want at up to what? X value is half of the total area under the curve. So, um, just by looking, um so Okay, it's a total area under a curve is 44 so x. So the center of mass is gonna be where up where the total, um, area under the curve above and below. It is 22. So, um, that if we we know going up to exceed six the area of the service 20 so and going up to seven, um, the area under the curve A's, um, you're gonna be Yes, we know that, um, it's gonna be between six. And seven. So now, um, we're gonna So yes. So now we're gonna get the ratio between six. And seven. So, um, we know, Yeah.

Let's start with party from from the graph, we can see the graph. The area under the curve is divided into little cubes, and we can try to find the area of each of these cubes. So for each cube, its height is one grams per centimeter. What's its length is one centimeter. So the area off this cube represents one grand. Okay, now, to find the mess off the rod from 0 to 5, we need to count how many of these cubes there are. So when you look at the graph, you can count 13 whole cubes, plus three halves off a cube. Now, one thing to notice. Two halves make one cube, so this turns out to be three cubes plus two halves, plus 1/2 this two halves. Make one cube. So have 13 cube plus one Q plus 1/2 and so then we have 14 cubes, plus 1/2 of a cube. So then, since we have since we know that one Cube represents one gram, we know that this is 14 grams and we have half a gram. So it's 0.5 Gramps. So that so then the mass of the Rod from 0 to 5 is 14.5 grams. Next we have Oops, Not not zero. We have 5 to 10. We can count that There are 29 cubes from 5 to 10. Plus, we have 1/2 of a cube. So substituting what we know about each cube, you know, this represents 29 grams plus 0.5 grams, so get 29.5 grams. Now, the mass of the entire rod should be the mass off the length 0 to 5, plus the mass off the length 5 to 10. So the total mass should be equal to 14 plus 14.5 plus 29.5 and you should get 44 grams in order to find the sent off mass off our the location off our center of mass on the road. Let's find what half of out mess is. So since our total mess from what we calculated before is 44 then our sent up then that half of this mass in 1/2 mess should be 22. Now what we want to do is now we want to find a point between zero and 10 such that both sides have 22 grams or in terms of the graph, we have 22 squares and when you look at look at the graph, you can't count. Roughly the point of 22 squares should be between between six and seven between six and seven, you can count roughly. Both sides has 22 grams slash square groups, number 22 Gramps, and that's it between six centimeters and seven centimeters.

And we have you on the road of negligible heat capacity and it is told that the temperature of one hand is maintain a juror degree Celsius and other end is slowly and linearly varied from 0 to 60 degree in 10 minutes. And we have to find the totally transmitted through this road in 10 minutes. So fast it'll fall. I am writing the given data for the institution. We have given the length of road that is led equals to 27 or I can say 0.2 m Area of the cross section of the road that is as equals to one semi square. And this will be one into 10 to the power minus four m square and thermal conductivity of the material off road. That? S case given to us which is 200 what permitted degree says Yes, it is told that the temperature of one end of the road is Increased uniformly by 60°C within 10 minutes. This means that the rate of increase of temperature of one end is 0.1°C per end. So this can be authored shown by that digital by Did it even B equals two. This is 16 upon 10 minutes. Or I can turn into 60 seconds. So this will come out to be 0.1 degree per second. This will tell that the temperature, The rate of increase of temperature of one end is 0.1°C in 1/2. So total heat flow can be found by adding heat flow every second. So total heat flow is given by that is Jeanette Will be equals two heat flowing each second. Time intervals that is submission of gay into D upon sorry, get into area upon day into temperature difference two to minus T. One into delta T. And for each interval, delta tease one. So this will come out. Unit is equals two K. A Body is the constant. So it will come out and this will come to be 0.1 plus 0.2 plus 0.3 plus. So on that test this will come up to 16. No some of anthems. Often A P. Is given by that is Some music was to end by two into first time plus last time. So apply this year. So this will be June it is equals two. This is came to europe on the beach and these This is 60 upon to into 0.1 Plus 60. Sorry this is 600. Now put the value of all the time the situation. So this is 202. 1 and 2. 10 to the power minus four upon This is 20 to 10 to the power -2 Into 300. This will the 60 point one. So after solving this, we get to net that will be 1800. After solving This will this will be 1800 jewel. So this is the answer for this hour given to shin. Thank you.