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Consider a solid sphere and a solid disk with the same radius and the same mass. Explain why the solid disk has a greater moment of inertia than the solid sphere, even though it has the same overall mass and radius.

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ajeigbeibraheem

Answer:

Explanation:

In a Solid sphere; the moment of inertia around its geometrical axis can be expressed by using the formula:

[tex]\mathtt{I_s = \dfrac{2}{5} M_s R^2_s}[/tex]

For the solid disk; the moment of inertia around the central axis is:

[tex]\mathtt{I_D= \dfrac{1}{2}M_DR_D^2}[/tex]

Suppose [tex]M_D = M_S[/tex]; then we can say both to be equal to M

As well as [tex]R_D = R_S[/tex]; then that too can be equal to R

Now;

[tex]\mathtt{I_s = \dfrac{2}{5} M R^2} --- (1)[/tex]

[tex]\mathtt{I_D= \dfrac{1}{2}MR^2}---(2)[/tex]

Multiplying equation (1) by 2, followed by dividing it by 2; we have:

[tex]\mathtt{I_s= \dfrac{2}{5}MR^2} \times \dfrac{2}{2}[/tex]

[tex]I_s = \dfrac{4}{5} \times \dfrac{1}{2}MR^2 \\ \\ I_s = \dfrac{4}{5}\times I_D \\ \\ I_s > I_D[/tex]

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