Moment of Inertia, Parallel Axes and Perpendicular Axes Theorems

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The moment of inertia of a particle of mass m about an axis A-A is defined as \begin{align} I=mr^2, \nonumber \end{align} where r is the perpendicular distance of the particle from the axis A-A.

Moment of Inertia of a Particle

The moment of inertia of a system of particles is given by \begin{align} I=\sum_i m_i r_i^2. \nonumber \end{align}

The moment of inertia of a body having continuous mass distribution is given by \begin{align} I=\int_\text{body} r^2\,\mathrm{d}m, \nonumber \end{align} where $r$ is perpendicular distance of the mass element $\mathrm{d}m$ from the axis. The integration is carried out over the entire body.

Moment of Inertia of Common Shapes

Moment of Inertia of Common Shapes
Moment of Inertia of Common Shapes

Theorem of Parallel Axes

Let $I_\mathrm{cm}$ be moment of inertia of a body of mass m about an axis passing through its centre of mass C. The moment of inertia of this body about a parallel axis at a perpendicular distance d from C is given by \begin{align} I_{\parallel}=I_\mathrm{cm}+md^2. \nonumber \end{align} The moment of inertia is the minimum for an axis passing through the centre of mass i.e., $ I_\mathrm{cm} \leq I_\parallel$.

Theorem of Parallel Axes

Theorem of Perpendicular Axes

Let a plane lamina lies in the x-y plane. The moment of inertia of the lamina about an axis perpendicular to its plane is given by \begin{align} I_z=I_x+I_y, \nonumber \end{align} where $I_x$ and $I_y$ are its moment of inertia about the x and y axes.

Theorem of Perpendicular Axes

Radius of Gyration

The radius of gyration ($k$) of a body of mass $m$ about an axis A-A is defined as \begin{align} k=\sqrt{I/m}, \nonumber \end{align} where $I$ is moment of inertia of the body about the axis A-A.

Solved Problems from IIT JEE

Problem from IIT JEE 2005

From a circular disc of radius $R$ and mass $9M$, a small disc of radius $R/3$ is removed. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through $O$ is,

Moment of Inertia Problem from IIT JEE 2005
  1. $4MR^2$
  2. $\frac{40}{9}MR^2$
  3. $10MR^2$
  4. $\frac{37}{9}MR^2$

Solution: The moment of inertia of disc of mass $9M$ and radius $R$ about an axis perpendicular to its plane and passing through its centre O is, \begin{align} I_\text{total}&=\frac{1}{2}(9M)R^2 \\ &=\frac{9}{2}MR^2.\nonumber \end{align} The mass of removed disc is $\frac{9M}{\pi R^2}\frac{\pi R^2}{9}=M$. The parallel axis theorem gives moment of inertia of the removed disc about axis passing through $O$ as, \begin{align} I_\text{removed}&=\frac{1}{2}M\left(\tfrac{R}{3}\right)^{\!2}+Md^2 \\ &=\tfrac{1}{18}MR^2+M\left(\tfrac{2R}{3}\right)^{\!2} \\ &=\tfrac{1}{2}MR^2.\nonumber \end{align} Using, $I_\text{total}=I_\text{remaining}+I_\text{removed}$, we get $I_\text{remaining}=4MR^2$.

Problem from IIT JEE 1997

A symmetric lamina of mass $M$ consists of a square shape with a semicircular section over each of the edge of the square as shown in figure. The side of the square is $2a$. The moment of inertia of the lamina about an axis through its centre of mass and perpendicular to the plane is $1.6 M a^2$. The moment of inertia of the lamina about the tangent AB in the plane of the lamina is _____.

Moment of Inertia Problem from IIT JEE 1997

Solution: Let $\mathrm{ZZ^\prime}$ be an axis perpendicular to the lamina and passing through its centre of mass O. Given $I_\mathrm{ZZ^\prime}=1.6Ma^2$.

Solution of Moment of Inertia Problem from IIT JEE 1997
By theorem of perpendicular axes, \begin{align} \label{rob:eqn:1} I_\mathrm{XX^\prime}+I_\mathrm{YY^\prime}=I_{ZZ^\prime}. \end{align} By symmetry, $I_\mathrm{XX^\prime}=I_\mathrm{YY^\prime}$. Substitute in above equation to get \begin{align} I_\mathrm{XX^\prime}&=\frac{1}{2}I_\mathrm{ZZ^\prime} \\ &=0.8Ma^2. \end{align} The theorem of parallel axes gives, \begin{align} I_\mathrm{AB}&=I_\mathrm{XX^\prime}+Md^2 \\ &=I_\mathrm{XX^\prime}+M(2a)^2 \\ &=4.8Ma^2.\nonumber \end{align}

More Solved Problems on Moment of Inertia from IIT JEE

Related Topics

References

  1. IIT JEE Physics by Jitender Singh and Shraddhesh Chaturvedi
  2. 300 Solved Problems on Rotational Mechanics by Jitender Singh and Shraddhesh Chaturvedi