Rotational Dynamics

1. Introduction

When a rigid body rotates about a fixed axis, every particle of the body moves in a circle whose centre lies on that axis. The whole body shares one angular displacement, one angular velocity, and one angular acceleration, even though different particles move with different linear speeds. Rotational dynamics studies how such rotation is produced and changed by torques, in exact parallel with how force changes linear motion.

The central quantity of rotation is the moment of inertia, which plays the same role in rotation that mass plays in linear motion. This chapter recalls the equations of angular motion, defines moment of inertia and radius of gyration, derives the moment of inertia of a thin rod, establishes the torque - angular-acceleration relation, treats work and power in rotation, and proves the conservation of angular momentum.

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2. Equations of Angular Motion (compared with linear motion)

For rotation with constant angular acceleration $\alpha$, the angular equations of motion have exactly the same form as the linear (kinematic) equations, with the linear quantities replaced by their angular counterparts.

The three equations of angular motion are:

$$ \begin{aligned} \omega &= \omega_0 + \alpha t \ \theta &= \omega_0 t + \tfrac{1}{2}\alpha t^2 \ \omega^2 &= \omega_0^2 + 2\alpha\theta \end{aligned} $$

These follow from the linear equations by the substitutions $s\to\theta$, $u\to\omega_0$, $v\to\omega$, $a\to\alpha$. The linear and angular quantities are connected through the radius $r$ of the circular path:

$$ \begin{aligned} s &= r\theta \ \qquad v &= r\omega \ \qquad a &= r\alpha \end{aligned} $$

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3. Moment of Inertia

The inertia of a body is its inability to change its state of rest or of uniform motion by itself. The rotational analogue of this is the moment of inertia (or rotational inertia): the property of a body by virtue of which it opposes any change in its state of rotational motion.

For a single particle of mass $m$ at a perpendicular distance $r$ from the axis of rotation, the moment of inertia is defined as:

$$I = mr^2$$

For a rigid body made of many particles of masses $m_1, m_2, m_3, \dots$ at perpendicular distances $r_1, r_2, r_3, \dots$ from the axis, the total moment of inertia is the sum of the individual contributions:

$$ \begin{aligned} I &= m_1 r_1^2 + m_2 r_2^2 + m_3 r_3^2 + \cdots \ &= \sum_i m_i r_i^2 \end{aligned} $$

Its SI unit is $\text{kg m}^2$ and its dimensional formula is $[\text{M L}^2]$.

Unlike mass, the moment of inertia of a body is not a fixed number: it depends on the mass of the body, on how that mass is distributed, and on the position and direction of the axis of rotation.

3.1 Radius of gyration

The radius of gyration $k$ of a body about an axis is the perpendicular distance from that axis at which the whole mass $M$ of the body could be concentrated so as to give the same moment of inertia:

$$ \begin{aligned} I &= M k^2 \quad\Rightarrow\quad k \ &= \sqrt{\frac{I}{M}} \end{aligned} $$

Its SI unit is the metre ($\text{m}$). The radius of gyration is a convenient single measure of how the mass of a body is spread out about the axis.

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