The temperature within the Earth’s crust increases about 1.0 C...

Question

The temperature within the Earth’s crust increases about 1.0 C° for each 30 m of depth. The thermal conductivity of the crust is 0.80 J/s C° .

(a) Determine the heat transferred from the interior to the surface for the entire Earth in 1.0 h.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. If the temperature within a planet's crust rises by 1.0 °C for every 32 minutes, calculate the heat transferred from the interior to the surface of the planet in one hour. Assume that the radius of the planet is 6.7 multiplied by 10 to the power of 6 m. And the thermal conductivity of the planet's crust is 0.82 joules per second multiplied by meters multiplied by degrees Celsius. Awesome. So with that in mind, we need to note that our final answer that we're ultimately trying to solve for is we're trying to figure out what the heat transferred value from the interior to the surface of the planet is in one hour. And that's our final value that we're ultimately trying to solve for is what is the heat transferred value? Awesome. So looking at our multiple choice answers, we could see that they're all in the same units of jewels. So let's read them off to see what our final answer might be. So A is 5.2 multiplied by 10 to the power of 16. B is 2.3 multiplied by 10 to the power of 17 C is 4.3 multiplied by 10 to the power of 18. And D is 5.7 multiplied by 10 to the power of 18. OK. So first off, let us write down all of our known variables. So we know that the thermal conductivity K is equal to 0.82 and its units are joules per second multiplied by meters multiplied by degrees Celsius. Awesome. So we need to say jewels per so jewels per second multiplied by meters multiplied by degrees Celsius. OK. So we need, we also know the temperature difference. So we know that the temperature distance delta T is equal to 1.0 °C. We also know that the depth which we're gonna call lowercase L is equal to 32 m. We also know the time T or the time T is equal to one hour. And we know that in one hour there is 3000 seconds in one hour. And this is using dimensional analysis which means that one hour is equal to 3600 seconds. We also know that the radius of the planet which I'm gonna call RP. So capital RP is equal to 6.7 multiplied by 10 to the power of 6 m. Awesome. So now we need to recall and use the equation to help us calculate the heat transfer rate, which states that Q divided by T is equal to K multiplied by capital, a multiplied by delta T divided by L Awesome. So now we need to recall and use the equation to help us calculate the total heat transferred Q. So we could get this equation by rearranging the heat transfer equation to isolate and solve for Q. So when we do that, we will find that Q is equal to K multiplied by capital A multiplied by delta T divided by L multiplied by lowercase T. So now we need to recall that. And then in this case, a represents the area, but we need to recall and use the equation for the area of a sphere. So with that in mind, let's make a little side note here off to the side in blue that the area of a sphere is equal to four multiplied by pi multiplied by radius square. So now we need to plug in this value into our equation. So when we do that, we will get that Q, so we say Q is equal to K multiplied by four multiplied by pi multiplied by RP square multiplied by delta T divided by L multiplied by T. So now we could plug in all of our known variables to solve for Q our final answer. So Q is equal to zero point 82 jewels per seconds, multiplied by meters multiplied by degrees Celsius multiplied by four, multiplied by pi multiplied by 6.6 or sorry, 6.67 multiplied by 10 to the power of 6 m squared because that's our radius value squared multiplied by one °C divided by 32 m. Awesome. Ok. So when we plug that, oh, wait. Oh, we brought out one more. Now we need to multiply by 3600 seconds means we can't forget our value for tea. Awesome. So now when we plug it in that long equation into our calculator, we will find that Q is equal to 5.2 when we round to two to round to one decimal place or two significant figures times 10 to the power of 16 jewels. And that's it we've solved for this problem. Hooray. We did it. So looking at our multiple choice answers, the correct answer has to be the letter A 5.2 multiplied by 10 to the power of 16 jewels and that's it. Thank you so much for watching. Hopefully that helped and I can't wait to see in the next video. Bye.

Struggling with Physics?

The temperature within the Earth’s crust increases about 1.0 C° for each 30 m of depth. The thermal conductivity of the crust is 0.80 J/s C° .

(a) Determine the heat transferred from the interior to the surface for the entire Earth in 1.0 h.

Watch next

Heat Transfer practice set

Struggling with Physics?

The temperature within the Earth’s crust increases about 1.0 C° for each 30 m of depth. The thermal conductivity of the crust is 0.80 J/s C° .(a) Determine the heat transferred from the interior to the surface for the entire Earth in 1.0 h.

Watch next

Heat Transfer practice set

The temperature within the Earth’s crust increases about 1.0 C° for each 30 m of depth. The thermal conductivity of the crust is 0.80 J/s C° .

(a) Determine the heat transferred from the interior to the surface for the entire Earth in 1.0 h.