A uniform circular plate of radius 2 R that has a circular hole of radius R cut

Question

A uniform circular plate of radius 2R that has a circular hole of radius R cut out of it. The center C' of the smaller circle is a distance d = 0.52 Rfrom the center C of the larger circle, see the figure. What is the position of the center of mass of the plate on the x axis? (Assume C is the origin of the coordinate and C' is on the +x axis. Hint: Try subtraction.)

I tried to create a system of equations relating the position of the center of the mass on the x axis with the y-coordinate at y=0, giving me

x' = [Mx-mx]/(m')

x'= [M(0)-mx]/(M-m)

and ended up with x/3, or 0.52/3

giving me 0.173 R.

But this isn't the right answer, so will someone please help me? :)

Never mind, I just got the answer. Apparently it's -0.173R.

A uniform circular plate of radius 2R that has a circular hole of radius R cut out of it. The center C' of the smaller circle is a distance d = 0.52 R from the center C of the larger circle, see the figure. What is the position of the center of mass of the plate on the x axis? (Assume C is the origin of the coordinate and C' is on the +x axis. Hint: Try subtraction.) I tried to create a system of equations relating the position of the center of the mass on the x axis with the y-coordinate at y=0, giving me

Explanation / Answer

radius of the larger circular plate = 2R radius of the smaller circle = R the distance between C (center of the larger circle) and C' (center of the smaller circle) d = x = 0.52 R let mass of the larger circle region is m. let mass of the smaller circle region is m'. therefore, the centre of mass of the plate on the x-axis is xcm = (m x + m' x')/(m+m') where total mass M = m + m' xcm = (m x + m' x')/(M) the position of the center of mass of the plate on the x axis is x' = [M xcm - m x]/(m') x' = [M xcm - m x]/(M-m) ............. (1) but the y -coordinate of center of mass of the total circle is located at y = 0. so the x-coordinate of the entire circle is at xcm = 0 x' = [M (0) - m x]/(M-m) = [- m x]/(M-m) (since mass m = ?*volume of the circle) = [-??R2(0.52R)]/ [(??(2R)2)-(??R2)] = -(0.52R)/3 = -x/3 (or) x' = 0.173 R
let mass of the larger circle region is m. let mass of the smaller circle region is m'. therefore, the centre of mass of the plate on the x-axis is xcm = (m x + m' x')/(m+m') where total mass M = m + m' xcm = (m x + m' x')/(M) the position of the center of mass of the plate on the x axis is x' = [M xcm - m x]/(m') x' = [M xcm - m x]/(M-m) ............. (1) but the y -coordinate of center of mass of the total circle is located at y = 0. so the x-coordinate of the entire circle is at xcm = 0 x' = [M (0) - m x]/(M-m) = [- m x]/(M-m) (since mass m = ?*volume of the circle) = [-??R2(0.52R)]/ [(??(2R)2)-(??R2)] = -(0.52R)/3 = -x/3 (or) x' = 0.173 R
x' = [M (0) - m x]/(M-m) = [- m x]/(M-m) (since mass m = ?*volume of the circle) = [-??R2(0.52R)]/ [(??(2R)2)-(??R2)] = -(0.52R)/3 = -x/3 (or) x' = 0.173 R

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