Question 39 of 280

If a solid sphere of mass $5kg$ and a disc of mass $4kg$ have the same radius, then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be $\dfrac{x}{7}$. The value of $x$ is ______.

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Answer: 5

Moment of inertia of the solid sphere about any diameter is given by, $\dfrac{2{m}_{1}{R}^{2}}{5}$.
Using parallel axis theorem, we get
${I}_{\text{solid sphere}}=\dfrac{2}{5}{m}_{1}{R}^{2}+{m}_{1}{R}^{2} =\dfrac{7}{5}{m}_{1}{R}^{2}$
$=\dfrac{7}{5}\times 5{R}^{2}$
$=7{R}^{2}$

Moment of inertia of the disc about the diameter is given by, $\dfrac{{m}_{2}{R}^{2}}{4}$.
Now using parallel axis theorem, we get
${I}_{\text{disc}}=\dfrac{{m}_{2}{R}^{2}}{4}+{m}_{2}{R}^{2} =\dfrac{5}{4}{m}_{2}{R}^{2}$
$=\dfrac{5}{4}\times 4{R}^{2}$
$=5{R}^{2}$
Therefore,$\dfrac{{I}_{\text{disc}}}{{I}_{\text{soild sphere}}}=\dfrac{5}{7}$
Thus, $x=5$.