Torque Knowledge-Based Study

Study notes

1. Torque is the rotational analog of linear force. While a net force causes an object to experience linear acceleration, a net torque causes an object to undergo angular acceleration about an axis of rotation.

   The effectiveness of a force in producing rotation depends not only on the magnitude of the force but also on where and in which direction the force is applied relative to the pivot point or rotation axis.

   Consider a force $\vec{F}$ acting on a rigid body at a position vector $\vec{r}$ relative to the chosen axis of rotation. The torque $\vec{\tau}$ is defined mathematically as the vector cross product:

   $$\vec{\tau} = \vec{r} \times \vec{F}$$

   The magnitude of the torque is given by:

   $$\tau = r F \sin\theta$$

   Here, $r$ is the distance from the rotation axis to the point of application of the force, $F$ is the magnitude of the applied force, and $\theta$ is the angle between the position vector $\vec{r}$ and the force vector $\vec{F}$ when placed tail-to-tail.

   The SI unit of torque is the newton-meter (N$\cdot$m).

   An equivalent and often convenient method to calculate the magnitude of torque uses the concept of the lever arm (or moment arm), denoted as $r_\perp$.

   The line of action of a force is the infinite line extending along the direction of the force vector. The lever arm is defined as the perpendicular distance from the axis of rotation to the line of action of the force:

   $$r_\perp = r \sin\theta$$

   Using the lever arm, the magnitude of torque is simply:

   $$\tau = r_\perp F$$

   When the force is directed along the line passing through the rotation axis ($\theta = 0^\circ$ or $\theta = 180^\circ$), the lever arm is zero, resulting in zero torque.

   Because torque is a vector quantity, its direction is perpendicular to the plane defined by $\vec{r}$ and $\vec{F}$, determined by the right-hand rule:

   • Point the fingers of your right hand along the position vector $\vec{r}$.
   • Curl your fingers toward the direction of the force vector $\vec{F}$.
   • Your outstretched thumb points in the direction of the torque vector $\vec{\tau}$.

   For two-dimensional planar problems, torque is generally assigned a scalar sign convention:

   • Torques causing counterclockwise (CCW) rotation are considered positive (directed out of the page).
   • Torques causing clockwise (CW) rotation are considered negative (directed into the page).

   When multiple forces act on a body, the net torque about a specific axis is the vector sum of all individual torques:

   $$\vec{\tau}_{\text{net}} = \sum \vec{\tau}_i$$

   According to Newton's second law for rotation, the net external torque acting on a rigid body with moment of inertia $I$ about a fixed axis produces an angular acceleration $\alpha$:

   $$\tau_{\text{net}} = I \alpha$$

   For a body to be in rotational equilibrium, the net torque acting on it must be zero ($\sum \tau = 0$).