## Related questions with answers

A solid cylinder with a radius of 4.0 cm has the same mass as a solid sphere of radius R. If the cylinder and sphere have the same moment of inertia about their centers, what is the sphere's radius?

Solution

Verified$\text{\underline{\textbf{Given values:}}}$

$2r=4 \: \text{cm}$

$r=2\: \text{cm}=0.02 \: \text{m}$

The moment of inertia of the cylinder is equal to:

$\begin{align*} I_1=\frac{1}{2}m_1{r_1}^2 \tag{Equation $1.$} \end{align*}$

The moment of inertia of the sphere is equal:

$\begin{align*} I_2=\frac{2}{5}m_2{r_2}^2 \tag{Equation $2.$} \end{align*}$

As we know that the moments of the cylinder and the sphere are equal, the mass of the cylinder and the sphere is also equal by equalizing equations $1$ and $2$, we find the diameter of the sphere:

$\begin{align*} I_1&=I_2\\ m_1&=m_2\\ I&=\frac{1}{2}m{r_1}^2\\ I&=\frac{2}{5}m{r_2}^2\\ \frac{1}{2}m{r_1}^2&=\frac{2}{5}m{r_2}^2\\ \frac{5}{4}{r_1}^2&={r_2}^2\\ r_2&=\sqrt {{r_1}^2 \frac{5}{4} }\\ r_2&=\frac{r_1}{2}\sqrt {5}\\ r_2&=0.02236067 \: \text{m}\\\\ \implies 2r_2&=\boxed{0.0447 \: \text{m}}\\ \end{align*}$

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