Newton's Laws of
Motion for objects moving in definable straightline
directions have been the basis for most of what we know about
the nature and actions of forces. In a very analogous way, we
can define equivalent ideas about the nature of rotating
objects, relative to some primary rotation axis.
When we deal with rotation, we talk about things like
rotation angle instead of distance and angular velocity in
place of (straightline) velocity. In fact, we can set out a
table of equivalent concepts between straightline motion and
rotation:
Linear Motion

Rotational Motion

Position

$$x$$

Velocity

$$v$$

Acceleration

$$a$$

Motion equations

\[
\left\{\begin{eqnarray*}
x &=& x_o + v_o t + \small{\frac{1}{2}} a\,t^2 \\
v &=& v_0 + at \\
v &=& \frac{\mathrm{d} x}{\mathrm{d} t} \\
a &=& \frac{\mathrm{d} v}{\mathrm{d} t}
\end{eqnarray*}\right.
\]

Mass (linear inertia)

$$m$$

Newton's 2^{nd} Law 
$$F = ma$$

Momentum

$$p=mv$$

Work

$$W=Fd$$

Kinetic energy

$$K=\frac{1}{2}mv^2$$

Power

$$P=Fv$$


$$θ$$

Angular position

$$ω$$

Angular velocity

$$α$$

Angular acceleration

\[
\left.\begin{eqnarray*}
θ &=& θ_o + ω_o t +
\small{\frac{1}{2}} α\,t^2 \\
ω &=& ω_o + α t \\
ω &=& \frac{\mathrm{d}θ}{\mathrm{d}t} \\
α &=& \frac{\mathrm{d}ω}{\mathrm{d}t}
\end{eqnarray*}\right\}
\]

Motion equations

$$I$$ 
Moment of inertia

$$τ = Iα$$

Newton's 2^{nd} Law

$$L = Iω$$

Angular momentum

$$W = τ\,θ$$

Work

$$K = \frac{1}{2}Iω^2$$ 
Kinetic energy

$$P = τ\,ω$$

Power


Some of the items in this table (like angular position or
acceleration) are straightforward to understand, others take a
bit more explanation. For example, the rotational equivalent of
mass is moment of inertia, $\small{I}$. In rotation, it is
just not how much matter (mass) you have that counts  but
where it is relative to the rotation axis. As an example,
think about why doorknobs are always on the edge of the door
away from the hinges. With doorknobs in their regular
locations, opening or closing a door is an easy matter. You
just apply a little force and the door begins to move. Next
time, though, try pushing with your usual doorclosing force on
a door  near to its hinges. You may be surprised on how much
harder you have to push to get the same result!
For an object whose mass, $m$, is concentrated at a
single point located a distance $r$ from the axis of
rotation, the equivalent moment of inertia is $\small{I = mr^2}$.
When the mass $m$ of an object is distributed through a region
of space, we can think of it as being made up of a collection
of very small point masses, $dm_i$, each located a unique distance
$r_i$ from the rotation axis with a moment of inertia
$\small{I_i\ = dm_i\,r_i^2}$. In this case, the total moment of
inertia $\small{I}$ is the sum of all the contributions $\small{I_i}$
from the little masses $dm_i$.
We will use our equivalence table to write the rotational
motion equivalents for Newton's laws of motion.
Continue to Rotational
Statements of Newton's laws.
©20032016
4Physics^{®}
