1. A uniform rod of mass M and length L can turn in a horizontal plane about a f

Question

1. A uniform rod of mass M and length L can turn in a horizontal plane about a frictionless vertical axle is threaded onto the rod and is held in place attached to one end of the rod. A bead of mass m = 3 a distance d = from the axle while the rod is made to spin at angular velocity wo. The bead is then released so that it can slide frictionlessly outward along the rod. What is the angular velocity of the rod when the bead reaches the end of the rod? How fast is the bead moving along the rod when it reaches the end? 3

Explanation / Answer

The intial moment of inertia of the system:

Ii = Irod + I bead

Ii = 1/3 M L^2 + M/3 (L/3)^2 = M L^2(1/3 + 1/27) = 10/27 M L^2

If = 1/3 M L^2 + M/3 L^2 = 2/3 M L^2

we know from conservation of angular momentum

Li = Lf

Ii w0 = If wf

wf = Ii w0/If

wf = (10/27 M L^2 x w0/2/3 M L^2) = 5/9 w0

Hence, wf = 5/9 w0

vf = 5/9 w0 x L

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