A set of four identical beads with mass m are glued onto a stiff circular wire w

Question

A set of four identical beads with mass m are glued onto a stiff circular wire with radius r (suppose the beads are at location 0, pi/2, pi, and 3pi/2, location is only for drawing the diagram, dont need these values for the question), the set can rotate around an axis through the centre, perpendicular to the plane of the drawing. express your answers in terms of m and r.

a) what is the moment of inertia, assuming the wire has no mass?

b) what is the moment of inertia, assuming the circular wire has mass m/2?

One of the beads falls off for the remainder of this question (for c-d)

c) what is the distance of the center of mass from the center of the circle, expressed as a fraction of r, assuming the circular wire has no mass?

d) what is the distance of the center of mass from the center of the circle, expressed as a fraction of r, assuming the circular wire has mass m/2?

Explanation / Answer

a)
I = m*r^2 + m*r^2 + m*r^2 + m*r^2
= 4*m*r^2

b)
Itotal = T of beads + I ring
I beads = 4*m*r^2
I ring = M*R^2 = (m/2)*r^2 = 0.5*m*r^2

Itotal = 4*m*r^2 + 0.5*m*r^2
= 4.5*m*r^2

c)
Since figure is symmetric, centre of mass will be at centre of circle

d)
Since figure is symmetric, centre of mass will be at centre of circle

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