Question:

Four holes of radius $R$ are cut from a thin square plate of side $4R$ and mass $M$. The moment of inertia of the remaining portion about z-axis is

Updated On: Jul 28, 2022
  • $\frac{\pi}{12}MR^{2}$
  • $\left(\frac{4}{3}-\frac{\pi}{4}\right)MR^{2}$
  • $\left(\frac{4}{3}-\frac{\pi}{6}\right)MR^{2}$
  • $\left(\frac{8}{3}-\frac{10\pi}{16}\right)MR^{2}$
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The Correct Option is D

Solution and Explanation

If M is mass of the square plate before cutting the holes, then mass of portion of each hole, $m=\frac{M}{16R^{2}}\times\pi R^{2}=\frac{M \pi}{16}$ $\therefore\,$ Moment of inertia of remaining portion about Z axis $I=I_{\text{square}}-4 I_{\text{hole}}$ $=\frac{M}{12}\left(16 R^{2}+16 R^{2}\right)-4\left[\frac{mR^{2}}{2}+m\left(\sqrt{2}R\right)^{2}\right]$ $=\frac{M}{12}\times32 R^{2}-10m R^{2}$ $=\frac{8}{3} MR^{2}-\frac{10}{16}MR^{2} \pi$ $I=\left(\frac{8}{3}-\frac{10\pi}{16}\right) MR^{2}.$
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Concepts Used:

System of Particles and Rotational Motion

  1. The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
  2. The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
  3. The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.