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My teacher taught us how to locate position of centre of mass in semicircular disc
She said circular disc is made up of two semicircles .It's centre of mass is located in centre .But When one of the semicircles is taken out from the circle , only one semicircle is left out.To locate the semicircular's centre of mass she taught us the following method
we will have to take help of calculus or of integration to be more precise .We should choose small semicircular ring of radius "p'and of thickness 'dp'.Total mass of that semicircular disc is given to be "m" and radius "r" so To detect position of centre of mass of semicircular disc we will use following formula
y coordinate of centre of mass=∫mass of that small semicircular ring multiplied by Y coordinate of it's centre of mass/∫dm(mass of that small semicircular ring )
Now mass of the small ring would be equal mass per unit area multiplied by area of ring i.e 2m/π r^2 multiplied by circumference by thickness i.e πp dp.Y coordinate of the ring can be treated as Y coordinate of it's centre of mass .In one of our derivations Y coordinate of semicircular ring came out to be 2 multiplied by it's radius/π
So y coordinate of centre of mass=∫2m/π r^2 multiplied by πp dp ×2pπ/ ∫dm an integration has upper and lower limit of r and zero respectively because we want to take all possible small rings of "0" to "r" radius.
=
So centre of mass of semicircular disc will be 4r/3 pi vertical distance from origin.I understood all these except how thickness comes out to be dp?How thickness can be derivative of radius(dR)?
She said circular disc is made up of two semicircles .It's centre of mass is located in centre .But When one of the semicircles is taken out from the circle , only one semicircle is left out.To locate the semicircular's centre of mass she taught us the following method
we will have to take help of calculus or of integration to be more precise .We should choose small semicircular ring of radius "p'and of thickness 'dp'.Total mass of that semicircular disc is given to be "m" and radius "r" so To detect position of centre of mass of semicircular disc we will use following formula
y coordinate of centre of mass=∫mass of that small semicircular ring multiplied by Y coordinate of it's centre of mass/∫dm(mass of that small semicircular ring )
Now mass of the small ring would be equal mass per unit area multiplied by area of ring i.e 2m/π r^2 multiplied by circumference by thickness i.e πp dp.Y coordinate of the ring can be treated as Y coordinate of it's centre of mass .In one of our derivations Y coordinate of semicircular ring came out to be 2 multiplied by it's radius/π
So y coordinate of centre of mass=∫2m/π r^2 multiplied by πp dp ×2pπ/ ∫dm an integration has upper and lower limit of r and zero respectively because we want to take all possible small rings of "0" to "r" radius.
=
So centre of mass of semicircular disc will be 4r/3 pi vertical distance from origin.I understood all these except how thickness comes out to be dp?How thickness can be derivative of radius(dR)?
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