Definition: Torque

The torque $\tau$ in rotation is equivalent to force $F$. We define $\tau$ relative to some chosen pivot point, from which $$\tau= Fr\sin\theta=F_{\perp} r=Fr_\perp$$ Here, $F$ is the force applied to the object, $r$ is the distance from the point of application to the pivot, and $\sin\theta$ is the angle between $F$ and $r$. We have $F_\perp$ as the perpendicular component of force relative to $r$, and $r_\perp$ to be the perpendicular component of distance relative to $F$. If an object is hinged, we choose the hinge of our object to be the pivot point.

Idea: Newton's Second Law

By using the substitutions $F\to \tau$, $m\to I$, and $a\to \alpha$, we can rewrite Newton's Second Law for rotational motion as $$\tau = I\alpha$$ Note that $\tau$, $I$, and $\alpha$ must be chosen to be calculated around the same pivot point for this equation to be valid.

Definition: Rotational Kinetic Energy

Rotational kinetic energy around the center of mass of an object is defined as $$K_{\text{rot}}=\frac 12 I\omega^2$$ and this is kinetic energy that is added to the translational kinetic energy $(1/2)mv^2$ of an object.

Idea: Statics

Static equilibrium problems ask you to analyze a situation where objects are perfectly at rest. For static equilibrium to occur, the net force and net torque acting on an object must be 0: $$\sum F=0\qquad \sum \tau = 0$$ Note that since $\tau=0$ about any pivot point, it is valuable to choose a pivot point that eliminates unknown forces (by placing the pivot on the point where the force is applied removes the torque contributions of this force).