Hamza M khan
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- Homework Statement
- The goal: finding the center of mas a semicircular wire/disk of on non negligible width, with the inner radius being R1 and out radius being R2.
My attempt:
1) Im gonna start this with a goal of setting up a reimann sum. First I divide the "arc"(?) of angle pi into n sub-arcs of equal angle Δθ
2) The total center of mass can be found if centers of mass of parts of the system are known. In each circular arc interval, I choose a height, Hi, approximating the height of the center os mass of each sub arc, hoping that the error goes to 0 in the limit as n goes to infinity, and multiply this by the mass of the sub arc. Pushing this through the limiting process, I set up the integral of H w.r.t m
3) finding Hi . Now, as Δθ goes to 0, the sector-difference region formed by each sub-arc should get closer and closer to a tilted rectangle. Assuming that to be true, the center of mass of each sub-arc( being approximated by a titled rectangle) would be a distance Hi=(R1+R2)sin(θ)/2 above the origin
4) lastly, since the shape has a constant mass per unit area, the differential mass and total mass can be replaced by differential area and total area. Using the sector area formula for each subinterval, the differntial area, dA, should be equal to 0.5dθ (R2^2-R1^2)
solving this gives me ycom=(R1+R2)/pi which upon looking up is clearly wrong. It is interesting thought that it gives the correct result when R1=R2 ( 0 thickness). What is the error in my reasoning?
- Relevant Equations
- Ycom=m1y1+m2y2+....miyi
My attempt:
1) I am going to start this with a goal of setting up a reimann sum. First I divide the "arc"(?) of angle pi into n sub-arcs of equal angle Δθ
2) The total center of mass can be found if centers of mass of parts of the system are known. In each circular arc interval, I choose a height, Hi, approximating the height of the center os mass of each sub arc, hoping that the error goes to 0 in the limit as n goes to infinity, and multiply this by the mass of the sub arc. Pushing this through the limiting process, I set up the integral of H w.r.t m
3) finding Hi . Now, as Δθ goes to 0, the sector-difference region formed by each sub-arc should get closer and closer to a tilted rectangle. Assuming that to be true, the center of mass of each sub-arc( being approximated by a titled rectangle) would be a distance Hi=(R1+R2)sin(θ)/2 above the origin
4) lastly, since the shape has a constant mass per unit area, the differential mass and total mass can be replaced by differential area and total area. Using the sector area formula for each subinterval, the differential area, dA, should be equal to 0.5dθ (R2^2-R1^2)
solving this gives me ycom=(R1+R2)/pi which upon looking up is clearly wrong. It is interesting thought that it gives the correct result when R1=R2 ( 0 thickness). What is the error in my reasoning?
1) I am going to start this with a goal of setting up a reimann sum. First I divide the "arc"(?) of angle pi into n sub-arcs of equal angle Δθ
2) The total center of mass can be found if centers of mass of parts of the system are known. In each circular arc interval, I choose a height, Hi, approximating the height of the center os mass of each sub arc, hoping that the error goes to 0 in the limit as n goes to infinity, and multiply this by the mass of the sub arc. Pushing this through the limiting process, I set up the integral of H w.r.t m
3) finding Hi . Now, as Δθ goes to 0, the sector-difference region formed by each sub-arc should get closer and closer to a tilted rectangle. Assuming that to be true, the center of mass of each sub-arc( being approximated by a titled rectangle) would be a distance Hi=(R1+R2)sin(θ)/2 above the origin
4) lastly, since the shape has a constant mass per unit area, the differential mass and total mass can be replaced by differential area and total area. Using the sector area formula for each subinterval, the differential area, dA, should be equal to 0.5dθ (R2^2-R1^2)
solving this gives me ycom=(R1+R2)/pi which upon looking up is clearly wrong. It is interesting thought that it gives the correct result when R1=R2 ( 0 thickness). What is the error in my reasoning?
