How to solve this particular task about centroids
$begingroup$
Can anyone help me with 29.I know how to find centroids when one function is given but in this one I don't think that knowing only the function of circle or rectangle will help us.I found that there is something called additive centroids but I don't know how that one works here.Can anyone fully explain this task and additive centroids please?
calculus integration centroid
asked Dec 24 '18 at 9:44
Arif RustamovArif Rustamov
387
$endgroup$
add a comment |
$begingroup$
Can anyone help me with 29.I know how to find centroids when one function is given but in this one I don't think that knowing only the function of circle or rectangle will help us.I found that there is something called additive centroids but I don't know how that one works here.Can anyone fully explain this task and additive centroids please?
calculus integration centroid
asked Dec 24 '18 at 9:44
Arif RustamovArif Rustamov
387
$endgroup$
2
$begingroup$
Centroids are averages. Get the centroid for each piece and then take the average of their coordinates. By symmetry this amounts to taking the average of the $y$ values from each individual centroid.
$endgroup$
– John Douma
Dec 24 '18 at 9:52
$begingroup$
Ohh really...I did not know that, but where does it come from?I think there should be a proof for that
$endgroup$
– Arif Rustamov
Dec 24 '18 at 9:54
$begingroup$
Look into the derivation of the centroid. It extends from a weighted average of discrete points in space to an integral.
$endgroup$
– John Douma
Dec 24 '18 at 9:55
$begingroup$
Honestly,I could not understand your point...
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
$begingroup$
I just want a solid proof that average can be found when we talk about unit shapes like the one given above
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
add a comment |
$begingroup$
Can anyone help me with 29.I know how to find centroids when one function is given but in this one I don't think that knowing only the function of circle or rectangle will help us.I found that there is something called additive centroids but I don't know how that one works here.Can anyone fully explain this task and additive centroids please?
calculus integration centroid
asked Dec 24 '18 at 9:44
Arif RustamovArif Rustamov
387
$endgroup$
Can anyone help me with 29.I know how to find centroids when one function is given but in this one I don't think that knowing only the function of circle or rectangle will help us.I found that there is something called additive centroids but I don't know how that one works here.Can anyone fully explain this task and additive centroids please?
calculus integration centroid
calculus integration centroid
asked Dec 24 '18 at 9:44
Arif RustamovArif Rustamov
387
asked Dec 24 '18 at 9:44
Arif RustamovArif Rustamov
387
asked Dec 24 '18 at 9:44
Arif RustamovArif Rustamov
387
asked Dec 24 '18 at 9:44
Arif RustamovArif Rustamov
387
asked Dec 24 '18 at 9:44
Arif RustamovArif Rustamov
387
387
2
$begingroup$
Centroids are averages. Get the centroid for each piece and then take the average of their coordinates. By symmetry this amounts to taking the average of the $y$ values from each individual centroid.
$endgroup$
– John Douma
Dec 24 '18 at 9:52
$begingroup$
Ohh really...I did not know that, but where does it come from?I think there should be a proof for that
$endgroup$
– Arif Rustamov
Dec 24 '18 at 9:54
$begingroup$
Look into the derivation of the centroid. It extends from a weighted average of discrete points in space to an integral.
$endgroup$
– John Douma
Dec 24 '18 at 9:55
$begingroup$
Honestly,I could not understand your point...
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
$begingroup$
I just want a solid proof that average can be found when we talk about unit shapes like the one given above
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
add a comment |
2
$begingroup$
Centroids are averages. Get the centroid for each piece and then take the average of their coordinates. By symmetry this amounts to taking the average of the $y$ values from each individual centroid.
$endgroup$
– John Douma
Dec 24 '18 at 9:52
$begingroup$
Ohh really...I did not know that, but where does it come from?I think there should be a proof for that
$endgroup$
– Arif Rustamov
Dec 24 '18 at 9:54
$begingroup$
Look into the derivation of the centroid. It extends from a weighted average of discrete points in space to an integral.
$endgroup$
– John Douma
Dec 24 '18 at 9:55
$begingroup$
Honestly,I could not understand your point...
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
$begingroup$
I just want a solid proof that average can be found when we talk about unit shapes like the one given above
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
2
2
$begingroup$
Centroids are averages. Get the centroid for each piece and then take the average of their coordinates. By symmetry this amounts to taking the average of the $y$ values from each individual centroid.
$endgroup$
– John Douma
Dec 24 '18 at 9:52
$begingroup$
Centroids are averages. Get the centroid for each piece and then take the average of their coordinates. By symmetry this amounts to taking the average of the $y$ values from each individual centroid.
$endgroup$
– John Douma
Dec 24 '18 at 9:52
$begingroup$
Ohh really...I did not know that, but where does it come from?I think there should be a proof for that
$endgroup$
– Arif Rustamov
Dec 24 '18 at 9:54
$begingroup$
Ohh really...I did not know that, but where does it come from?I think there should be a proof for that
$endgroup$
– Arif Rustamov
Dec 24 '18 at 9:54
$begingroup$
Look into the derivation of the centroid. It extends from a weighted average of discrete points in space to an integral.
$endgroup$
– John Douma
Dec 24 '18 at 9:55
$begingroup$
Look into the derivation of the centroid. It extends from a weighted average of discrete points in space to an integral.
$endgroup$
– John Douma
Dec 24 '18 at 9:55
$begingroup$
Honestly,I could not understand your point...
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
$begingroup$
Honestly,I could not understand your point...
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
$begingroup$
I just want a solid proof that average can be found when we talk about unit shapes like the one given above
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
$begingroup$
I just want a solid proof that average can be found when we talk about unit shapes like the one given above
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As you can probably guess (by symmetry or intuition), the centroid of the rectangle is at $C_1 =(0,h/2)$. Now the centroid of semicircle is actually $frac{4r}{3pi}$ above the centre of the circle which in our case makes it
$$C_2 = (0,h+frac{4a}{3pi})$$
To calculate the centroid of the entire figure you simply use weighted averages
$$C = frac{C_1cdot 2ah + C_2cdotpi r^2}{2ah+pi r^2}$$
You can take it from here, I believe.
Reference - http://datagenetics.com/blog/january52017/index.html
answered Dec 24 '18 at 10:27
Sauhard SharmaSauhard Sharma
953318
$endgroup$
$begingroup$
Huge thanks for great answer.One more thing,it is the first time that I encounter term weighted averages, What does it mean in general?Can that formula be applied to all kind of united shapes?
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:38
$begingroup$
As you see I am reading this lesson from the book and there was nothing about this term
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:39
$begingroup$
A normal or non-weighted average is where you sum everything up and divide by the total number. However, in a weighted average, every quantity has a different weight and each value is multiplied by its weight and then divided by the sum of weights. Reference - math.tutorvista.com/statistics/weighted-average.html
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Also could you please accept the answer then?
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Got it finally.Huge thanks
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:50
add a comment |
Your Answer
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1 Answer
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1 Answer
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active
oldest
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oldest
votes
$begingroup$
As you can probably guess (by symmetry or intuition), the centroid of the rectangle is at $C_1 =(0,h/2)$. Now the centroid of semicircle is actually $frac{4r}{3pi}$ above the centre of the circle which in our case makes it
$$C_2 = (0,h+frac{4a}{3pi})$$
To calculate the centroid of the entire figure you simply use weighted averages
$$C = frac{C_1cdot 2ah + C_2cdotpi r^2}{2ah+pi r^2}$$
You can take it from here, I believe.
Reference - http://datagenetics.com/blog/january52017/index.html
answered Dec 24 '18 at 10:27
Sauhard SharmaSauhard Sharma
953318
$endgroup$
$begingroup$
Huge thanks for great answer.One more thing,it is the first time that I encounter term weighted averages, What does it mean in general?Can that formula be applied to all kind of united shapes?
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:38
$begingroup$
As you see I am reading this lesson from the book and there was nothing about this term
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:39
$begingroup$
A normal or non-weighted average is where you sum everything up and divide by the total number. However, in a weighted average, every quantity has a different weight and each value is multiplied by its weight and then divided by the sum of weights. Reference - math.tutorvista.com/statistics/weighted-average.html
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Also could you please accept the answer then?
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Got it finally.Huge thanks
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:50
add a comment |
$begingroup$
As you can probably guess (by symmetry or intuition), the centroid of the rectangle is at $C_1 =(0,h/2)$. Now the centroid of semicircle is actually $frac{4r}{3pi}$ above the centre of the circle which in our case makes it
$$C_2 = (0,h+frac{4a}{3pi})$$
To calculate the centroid of the entire figure you simply use weighted averages
$$C = frac{C_1cdot 2ah + C_2cdotpi r^2}{2ah+pi r^2}$$
You can take it from here, I believe.
Reference - http://datagenetics.com/blog/january52017/index.html
answered Dec 24 '18 at 10:27
Sauhard SharmaSauhard Sharma
953318
$endgroup$
$begingroup$
Huge thanks for great answer.One more thing,it is the first time that I encounter term weighted averages, What does it mean in general?Can that formula be applied to all kind of united shapes?
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:38
$begingroup$
As you see I am reading this lesson from the book and there was nothing about this term
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:39
$begingroup$
A normal or non-weighted average is where you sum everything up and divide by the total number. However, in a weighted average, every quantity has a different weight and each value is multiplied by its weight and then divided by the sum of weights. Reference - math.tutorvista.com/statistics/weighted-average.html
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Also could you please accept the answer then?
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Got it finally.Huge thanks
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:50
add a comment |
$begingroup$
As you can probably guess (by symmetry or intuition), the centroid of the rectangle is at $C_1 =(0,h/2)$. Now the centroid of semicircle is actually $frac{4r}{3pi}$ above the centre of the circle which in our case makes it
$$C_2 = (0,h+frac{4a}{3pi})$$
To calculate the centroid of the entire figure you simply use weighted averages
$$C = frac{C_1cdot 2ah + C_2cdotpi r^2}{2ah+pi r^2}$$
You can take it from here, I believe.
Reference - http://datagenetics.com/blog/january52017/index.html
answered Dec 24 '18 at 10:27
Sauhard SharmaSauhard Sharma
953318
$endgroup$
As you can probably guess (by symmetry or intuition), the centroid of the rectangle is at $C_1 =(0,h/2)$. Now the centroid of semicircle is actually $frac{4r}{3pi}$ above the centre of the circle which in our case makes it
$$C_2 = (0,h+frac{4a}{3pi})$$
To calculate the centroid of the entire figure you simply use weighted averages
$$C = frac{C_1cdot 2ah + C_2cdotpi r^2}{2ah+pi r^2}$$
You can take it from here, I believe.
Reference - http://datagenetics.com/blog/january52017/index.html
answered Dec 24 '18 at 10:27
Sauhard SharmaSauhard Sharma
953318
answered Dec 24 '18 at 10:27
Sauhard SharmaSauhard Sharma
953318
answered Dec 24 '18 at 10:27
Sauhard SharmaSauhard Sharma
953318
answered Dec 24 '18 at 10:27
Sauhard SharmaSauhard Sharma
953318
953318
$begingroup$
Huge thanks for great answer.One more thing,it is the first time that I encounter term weighted averages, What does it mean in general?Can that formula be applied to all kind of united shapes?
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:38
$begingroup$
As you see I am reading this lesson from the book and there was nothing about this term
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:39
$begingroup$
A normal or non-weighted average is where you sum everything up and divide by the total number. However, in a weighted average, every quantity has a different weight and each value is multiplied by its weight and then divided by the sum of weights. Reference - math.tutorvista.com/statistics/weighted-average.html
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Also could you please accept the answer then?
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Got it finally.Huge thanks
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:50
add a comment |
$begingroup$
Huge thanks for great answer.One more thing,it is the first time that I encounter term weighted averages, What does it mean in general?Can that formula be applied to all kind of united shapes?
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:38
$begingroup$
As you see I am reading this lesson from the book and there was nothing about this term
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:39
$begingroup$
A normal or non-weighted average is where you sum everything up and divide by the total number. However, in a weighted average, every quantity has a different weight and each value is multiplied by its weight and then divided by the sum of weights. Reference - math.tutorvista.com/statistics/weighted-average.html
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Also could you please accept the answer then?
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Got it finally.Huge thanks
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:50
$begingroup$
Huge thanks for great answer.One more thing,it is the first time that I encounter term weighted averages, What does it mean in general?Can that formula be applied to all kind of united shapes?
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:38
$begingroup$
Huge thanks for great answer.One more thing,it is the first time that I encounter term weighted averages, What does it mean in general?Can that formula be applied to all kind of united shapes?
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:38
$begingroup$
As you see I am reading this lesson from the book and there was nothing about this term
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:39
$begingroup$
As you see I am reading this lesson from the book and there was nothing about this term
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:39
$begingroup$
A normal or non-weighted average is where you sum everything up and divide by the total number. However, in a weighted average, every quantity has a different weight and each value is multiplied by its weight and then divided by the sum of weights. Reference - math.tutorvista.com/statistics/weighted-average.html
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
A normal or non-weighted average is where you sum everything up and divide by the total number. However, in a weighted average, every quantity has a different weight and each value is multiplied by its weight and then divided by the sum of weights. Reference - math.tutorvista.com/statistics/weighted-average.html
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Also could you please accept the answer then?
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Also could you please accept the answer then?
$endgroup$
– Sauhard Sharma
Dec 24 '18 at 10:45
$begingroup$
Got it finally.Huge thanks
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:50
$begingroup$
Got it finally.Huge thanks
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:50
add a comment |
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2
$begingroup$
Centroids are averages. Get the centroid for each piece and then take the average of their coordinates. By symmetry this amounts to taking the average of the $y$ values from each individual centroid.
$endgroup$
– John Douma
Dec 24 '18 at 9:52
$begingroup$
Ohh really...I did not know that, but where does it come from?I think there should be a proof for that
$endgroup$
– Arif Rustamov
Dec 24 '18 at 9:54
$begingroup$
Look into the derivation of the centroid. It extends from a weighted average of discrete points in space to an integral.
$endgroup$
– John Douma
Dec 24 '18 at 9:55
$begingroup$
Honestly,I could not understand your point...
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10
$begingroup$
I just want a solid proof that average can be found when we talk about unit shapes like the one given above
$endgroup$
– Arif Rustamov
Dec 24 '18 at 10:10