Why does it require less work?

jwxie · Mar 25, 2010

1. The problem statement, all variables and given/known data

Four tiny spheres are fastened to the ends of two rods of negligible mass lying in the xy plane.
figure is illustrated below.

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2. Relevant equations

3. The attempt at a solution
I see I can derive the problem using equations to answer my question.
But here is a statement the textbook made

In part A we are ask "if the system rotates about the y axis" whereas in part B, we are asked "rotate the system about z axis"

In this problem, we will expect all four spheres will be revolving when it is revolve about z axis.

"Based on the work-kinetic energy theorem, that the rotational kinetic energy in part A, is smaller than that in part b, which indicates it would require less work to set system into rotation about the y axis than about z axis"

How did the author come up with this statement using W-K theorem?

CompuChip · Mar 25, 2010

Let me assume that the whole construction is like a plus sign, with all spheres identical and at the same distance d to the centre.

Let I be the moment of inertia of such a sphere around the centre (I suppose you need the parallel axis theorem and it will be something like I = I_sphere + m d², but that's irrelevant). Then when you rotate around the y-axis, only two of the spheres are moving, so you will get something like
I_tot = 2I.
When you rotate around the z-axis, all four are moving and you have
I_tot = 4I.

Since the work-energy theorem states that the work W is proportional to I_tot, the conclusion would follow.

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Why does it require less work?

jwxie

CompuChip

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