Question
Asked Oct 9, 2019
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Two balls with masses M and m are connected by a rigid
rod of length L and negligible mass as shown in Figure
P10.28. For an axis perpendicular to the rod, (a) show that
the system has the minimum moment of inertia when the
axis passes through the center of mass. (b) Show that this R moment of inertia is I = uL2, where u=mM/(m 1 M).

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Step 1

 Given:

Masses of the balls are M and m respectively.

Length of the rod, = L

We know that moment of inertia around the axis:

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I = mr2

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Step 2

(a)

Writing the moment of inertia from the fig.

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I Mx +m(L-x) (i) Differentiating w.r.tx dI - 2Mr + 2m(L- х)x (0-1) dx (i) dl To calculate the minimum value, putting 0: dx 0 2Mx 2mL+2mx 2mL= 2(M +m)x mL x = (M +m Differentiating (ii) w.r.t x 3D 2М - 2т > 0 dx2 Therefore: mL х. min (M +m)

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Step 3

Which is also the center of...

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