A uniform rod of mass M and length d is initially at rest on a horizontal and fr

Question

A uniform rod of mass M and length d is initially at rest on a horizontal and frictionless table contained in the xy plane, the plane of the screen. The figure is a top view, gravity points to the screen. The rod is free to rotate about an axis perpendicular to the plane and passing through the pivot point at a distance d/3 measured from one of its ends as shown. A small point mass m, moving with speed vo, hits the rod and sticks to it at the point of impact at a distance d/3 from the pivot.

a. If the mass of the rod is M=4m, what is the magnitude of the angular velocity of the rod+small mass system after the collision?

b. Using again M=4m. What is the speed of the center of mass of the rod right after collision?

Explanation / Answer

Initial angular momentum relative to the pivot point: m v0 (d/3)

Final angular momentum: I w, with I comprised of three parts.
I1 the moment of inertia of a point mass m at distance d/3 from the center of rotation: m (d/3)^2
I2 the moment of inertia for a rod of mass M/3 and length d/3 about its end: 1/3 (M/3) (d/3)^2
I3 the moment of inertia for a rod of mass 2M/3 and length 2d/3: 1/3 (2M/3) (2d/3)^2

I = I1+I2+I3 = 1/9 (m+M) d^2

No external torques, so angular momentum is conserved=>
I w = m v0 d/3

w = m v0 d / (3I) = 3m/(m+M) * v0/d

With M=4m

w = 3/5 * v0/d

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