Question

Moment of inertia of a solid about its geometrical axis is given by $l=\frac{2}{5}M{R}^2$ where $M$ is mass & $R$ is radius. Find out the rate by which its moment of inertia is changing keeping density constant at the moment $R=1m,M=1kg$ & rate of change of radius w.r.t. time $2m{s}^{-1}$

• A. $4kgm{s}^{-1}$
• B. $2kg{m}^2{s}^{-1}$
• C. $4kg{m}^2{s}^{-1}$
• D. None of these

Answer 1

Answer: (C)

$4kg{m}^2{s}^{-1}$

$l=\frac{2}{5}M{R}^2=\frac{2}{5}\left(\begin{array}{c}\frac{4}{3}\pi R^3\rho \end{array}\right)R^2=\frac{8}{15}\pi \rho R^5$

$\frac{dl}{dt}=\left(\begin{array}{c}\frac{8}{15}\pi \rho \end{array}\right)\left(5R^4\right)\frac{dR}{dt}=\left(\begin{array}{c}\frac{8\pi }{15}\end{array}\right)\left(\begin{array}{c}\frac{M}{4/3\pi R^3}\end{array}\right)\left(5R^4\right)$

$\frac{dR}{dt}=2MR\left(\begin{array}{c}\frac{dR}{dt}\end{array}\right)=\left(2\right)\left(1\right)\left(1\right)\left(2\right)=4kg{m}^2{s}^{-1}$