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Newton's Laws for Rotation

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Contents:   ● Linear & Rotational Equiv.   ● Rotational Form of Newton's Laws   ● Torque & Precession

Newton's Laws of Motion for objects moving in definable straight-line 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 (straight-line) velocity. In fact, we can set out a table of equivalent concepts between straight-line 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 2nd 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 2nd 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 door-closing 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.

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