AP Physics 1 : Introduction: Rotational Statics & Dynamics Study Notes

Torque

Notes: torque $(\overrightarrow{\boldsymbol{\tau}})$ : a vector quantity that measures the effectiveness of a force in causing rotation. Take care to distinguish the Greek letter ” $\tau$ ” from the Roman letter ” $\mathrm{t}$ “. Torque is measured in units of newton-meters:
$
1 \mathrm{~N} \cdot \mathrm{m}=1 \frac{\mathrm{kg} \cdot \mathrm{m}^2}{\mathrm{~s}^2}
$
Note that work and energy (which we will study later) are also measured in newton-meters. However, work and energy are different quantities from torque, and are not interchangeable. (Among other differences, work and energy are scalar quantities, and torque is a vector quantity.)

axis of rotation: the point around which an object rotates.

fulcrum: the point around which a lever pivots. Also called the pivot.

lever arm: the distance from the axis of rotation that a force is applied, causing a torque.

Just as force is the quantity that causes linear acceleration, torque is the quantity that causes a change in the speed of rotation (rotational acceleration).
Because inertia is a property of mass, Newton’s second law is the relationship between force and inertia. Newton’s second law in rotational systems looks similar to Newton’s second law in linear systems:
$
\begin{array}{cc}
\overrightarrow{\boldsymbol{a}}=\frac{\sum \overrightarrow{\boldsymbol{F}}}{m}=\frac{\overrightarrow{\boldsymbol{F}}_{n e t}}{m} & \overrightarrow{\boldsymbol{\alpha}}=\frac{\sum \overrightarrow{\boldsymbol{\tau}}}{I}=\frac{\overrightarrow{\boldsymbol{\tau}}_{\text {net }}}{I} \\
\overrightarrow{\boldsymbol{F}}_{\text {net }}=m \overrightarrow{\boldsymbol{a}} & \overrightarrow{\boldsymbol{\tau}}_{\text {net }}=I \overrightarrow{\boldsymbol{\alpha}}^* \\
\text { linear } & \text { rotational }
\end{array}
$

As you should remember, a net force of zero, that means all forces cancel in all directions and there is no acceleration. If there is no acceleration $(\vec{a}=0)$, the velocity remains constant (which may or may not equal zero).
Similarly, if the net torque is zero, then the torques cancel in all directions and there is no angular acceleration. If there is no angular acceleration $(\vec{\alpha}=0)$, then the angular velocity remains constant (which may or may not equal zero).
rotational equilibrium: when all of the torques on an object cancel each other’s effects (resulting in a net force of zero) and the object either does not rotate or rotates with a constant angular velocity.
Torque is also the cross product of distance from the center of rotation (“lever arm”) $\times$ force:
$\overrightarrow{\mathbf{\tau}}=\overrightarrow{\mathbf{r}} \times \overrightarrow{\mathbf{F}} \quad$ which gives: $\quad\|\overrightarrow{\mathbf{\tau}}\|=\tau=r F \sin \theta=r F_{\perp}$
where $\theta$ is the angle between the lever arm and the applied force.
We use the variable $r$ for the lever arm (which is a distance) because torque causes rotation, and $r$ is the distance from the center of the circle (radius) at which the force is applied.

$F \sin \theta$ is sometimes written as $F_{\perp}$ (the component of the force that is perpendicular to the radius) and sometimes $F_{\|}$(the component of the force that is parallel to the direction of motion). These notes will use $F_{\perp}$, because in many cases the force is applied to a lever, and the component of the force that causes the torque is perpendicular to the lever itself, so it is easy to think of it as “the amount of force that is perpendicular to the lever”. This gives the equation:
$
\tau=r F_{\perp}
$

Of course, because torque is the cross product of two vectors, it is a vector whose direction is perpendicular to both the lever arm and the force.

This is an application of the “right hand rule.” If your fingers of your right hand curl from the first vector $(\overrightarrow{\boldsymbol{r}})$ to the second $(\overrightarrow{\boldsymbol{F}})$, then your thumb points in the direction of the resultant vector $(\overrightarrow{\boldsymbol{\tau}})$. Note that the direction of the torque vector is parallel to the axis of rotation.

Note, however, that you can’t “feel” torque; you can only “feel” force. Most people think of the “direction” of a torque as the direction of the rotation that the torque would produce (clockwise or counterclockwise). In fact, the College Board usually uses this convention.

Mathematically, the direction of the torque vector is needed only to give torques a positive or negative sign, so torques in the same direction add and torques in opposite directions subtract. In practice, most people find it easier to define the positive direction for rotation (clockwise $U$ or counterclockwise $U$ ) and use those for positive or negative torques in the problem, regardless of the direction of the torque vector.