Chapter 8, Problem 9Q

This book has three symmetry axes through its center, all mutually perpendicular. The book’s moment of inertia would be smallest about which of the three? Explain.

To determine

The axis of symmetry for the book about which the moment of inertia is the smallest.

Answer to Problem 9Q

Solution:

The moment of inertia for the book is the smallest along the axis passing through its center of mass and parallel to the length (longer side) of the book.

Explanation of Solution

Formula used:

Moment of inertia of rod:

  $I=\frac{1}{12}m{l}^2$

Where, m is the mass and l is the length.

According to perpendicular axis theorem:

  $I_z=I_x+I_y$

Calculation:

  

The three axes of symmetry for the book are shown above. Let us consider that the book’s center of mass is at the origin and the book’s mass is m . The moments of inertia along the three axes are then given by,

  $I_x=\frac{1}{12}m{l}^2$

  $I_y=\frac{1}{12}m{w}^2$

  $I_z=\frac{1}{12}m\left(l^2+w^2\right)$

For the book, the length (l) is greater than the width ( w ).

  $I_z>I_x>I_y$

Conclusion:

Therefore, the moment of inertia for the book is the smallest along the axis passing through its center of mass and parallel to the length (longer side) of the book.