Four beads each of mass MM are attached at various locations to a hoop of mass M

Question

Four beads each of mass MM are attached at various locations to a hoop of mass MM and radius RR (see figure below). Find the center of mass of the hoop and beads. Assume ?1?1 = 65° and ?2?2 = 42°.

1) What is the xx component in terms of RR? (Express your answer to two significant figures.)

2) What is the yy component in terms of RR? (Express your answer to two significant figures.)

Four beads each of mass M are attached at various locations to a hoop of mass M and radius R (see figure below). Find the center of mass of the hoop and beads. Assume 01 = 65° and O2 = 42°. M R Mo 350

Explanation / Answer

Position of the bead in the top left quadrant, (x1, y1) = (- R cos(35), R sin(35))
(x1, y1) = ( - 0.819 R, 0.574 R)

Position of the bead in the top right quadrant, (x2, y2) = (R cos(42), R sin(42))
(x2, y2) = (0.743 R, 0.669 R)

Position of the bead in the bottom left quadrant, (x3, y3) = (- R cos(65), - R sin(65))
(x3, y3) = (- 0.423 R, - 0.906 R)

Position of the bead in the bottom right quadrant, (x4, y4) = (R cos(50), - R sin(50))
(x4, y4) = (0.643 R, - 0.766 R)

Position of center of mass of the hoop, (x5, y5) = (0, 0)

Center of mass of the structure, Xcm = [x1M + x2M + x3M + x4M + x5M] / [M + M + M + M + M]
= [- 0.819 R x M + 0.743 R x M + (- 0.423 R) x M + 0.643 R x M + 0 x M] / 5M
= 0.029 R
Similarly, Ycm = [y1M + y2M + y3M + y4M + y5M] / [M + M + M + M + M]
= [0.574 R x M + 0.669 R x M + (- 0.906 R) x M + (- 0.766 R) x M + 0 x M] / 5M
= - 0.086 R
So the center of mass of the structure is (0.029 R, - 0.086 R)

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