Rotation flash cards

1. 1.Define moment of inertia of a particle and of a system about an axis.

   For a particle, $I = mr^2$ where $r$ is the perpendicular distance from the axis. For a system, $I = \sum m_i r_i^2$, and for a continuous body $I = \int r^2\,dm$. It is the rotational analogue of mass.

   Hint: Sum of $mr^2$ about the axis.

2. 2.State the parallel axis theorem.

   $I = I_{cm} + Md^2$, where $I_{cm}$ is the moment of inertia about an axis through the centre of mass and $d$ is the perpendicular distance between the two parallel axes.

   Hint: Shift from CM by $Md^2$.

3. 3.State the perpendicular axis theorem and its restriction.

   For a planar (2D) lamina, $I_z = I_x + I_y$, where $x,y$ lie in the plane and $z$ is perpendicular to it. It applies only to flat laminae, not 3D bodies.

   Hint: Lamina only; $z$-axis is normal to the plane.

4. 4.Moment of inertia of a thin uniform rod (length $L$, mass $M$) about a perpendicular axis through its centre?

   $I = \frac{1}{12}ML^2$. By parallel axis theorem, about one end it becomes $I = \frac{1}{3}ML^2$.

   Hint: Centre gives twelfth; end gives third.

5. 5.Moment of inertia of a uniform ring (mass $M$, radius $R$) about its central axis and about a diameter?

   About the central (symmetry) axis: $I = MR^2$. About a diameter, by perpendicular axis theorem $2I_d = MR^2$, so $I_d = \frac{1}{2}MR^2$.

   Hint: All mass at radius $R$; diameter is half.

6. 6.Moment of inertia of a uniform disc (mass $M$, radius $R$) about its central axis and about a diameter?

   Central axis: $I = \frac{1}{2}MR^2$. About a diameter: $I_d = \frac{1}{4}MR^2$ (from perpendicular axis theorem).

   Hint: Half for axis, quarter for diameter.

7. 7.Moment of inertia of a solid sphere (mass $M$, radius $R$) about a diameter, and about a tangent?

   About a diameter: $I = \frac{2}{5}MR^2$. About a tangent: $I = \frac{2}{5}MR^2 + MR^2 = \frac{7}{5}MR^2$.

   Hint: Two-fifths; tangent adds $MR^2$.

8. 8.Moment of inertia of a thin hollow (spherical shell) sphere about a diameter and about a tangent?

   About a diameter: $I = \frac{2}{3}MR^2$. About a tangent: $I = \frac{2}{3}MR^2 + MR^2 = \frac{5}{3}MR^2$.

   Hint: Two-thirds for shell; tangent adds $MR^2$.

9. 9.Moment of inertia of a solid cylinder (mass $M$, radius $R$) about its own (longitudinal) axis?

   $I = \frac{1}{2}MR^2$, independent of length $L$. Same as a disc because it is a stack of discs about the common axis.

   Hint: Length doesn't matter for the long axis.

10. 10.Moment of inertia of a hollow cylinder / thin cylindrical shell about its longitudinal axis?

    $I = MR^2$ for a thin shell. For a thick hollow cylinder with radii $R_1, R_2$: $I = \frac{1}{2}M(R_1^2 + R_2^2)$.

    Hint: Thin shell behaves like a ring.

11. 11.Define radius of gyration $k$.

    $k$ is defined by $I = Mk^2$, so $k = \sqrt{I/M}$. It is the distance from the axis at which the whole mass could be concentrated to give the same moment of inertia.

    Hint: $I = Mk^2$.

12. 12.What is the radius of gyration of a solid sphere about its diameter?

    $I = \frac{2}{5}MR^2 = Mk^2 \Rightarrow k = \sqrt{\frac{2}{5}}\,R = R\sqrt{0.4} \approx 0.632R$.

    Hint: Take $\sqrt{2/5}$ of $R$.

13. 13.Moment of inertia of a uniform rectangular plate (sides $a,b$, mass $M$) about an axis through its centre perpendicular to the plate?

    $I = \frac{1}{12}M(a^2 + b^2)$, obtained by adding $\frac{1}{12}Ma^2$ and $\frac{1}{12}Mb^2$ via the perpendicular axis theorem.

    Hint: Add the two in-plane contributions.

14. 14.Define torque as a vector and give its magnitude.

    $\vec{\tau} = \vec{r}\times\vec{F}$, magnitude $\tau = rF\sin\theta = F\,d$, where $d = r\sin\theta$ is the moment arm (perpendicular distance from axis to line of action of the force).

    Hint: Force times perpendicular lever arm.

15. 15.State the rotational form of Newton's second law about a fixed axis.

    $\tau_{net} = I\alpha$, where $\alpha$ is the angular acceleration. It holds about a fixed axis or about the centre of mass even if the CM accelerates.

    Hint: Rotational $F = ma$.

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