Torque and Rotational Equilibrium: Study Notes

Torque and Rotational Motion

Definition and Physical Meaning of Torque

Torque is the rotational equivalent of force. It quantifies the tendency of a force to cause an object to rotate about an axis or pivot point.

• Force causes linear acceleration; torque causes angular acceleration.

• Torque measures the effectiveness of a force in producing rotation.

Factors Affecting Torque

The ability of a force to cause rotation depends on three main factors:

• Magnitude of the force ($F$)

• Distance from the pivot (center of rotation) to the point of force application ($r$)

• Angle at which the force is applied ($\theta$)

Mathematical Expression for Torque

Torque equals a force with a perpendicular component acting at a distance from the pivot:

• Formula:

$\tau = r F_\perp$

• Where $F_\perp$ is the component of force perpendicular to the lever arm.

• Torque has units of Newton-meters (N·m).

General Formula for Torque

When the force is applied at an angle $\theta$ to the lever arm:

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

• $r$ = distance from axis of rotation to the force

• $\theta$ = angle between the position vector $\vec{r}$ and the force vector $\vec{F}$

Examples of Torque

• Solid Rod Example: Applying force at the end of a rod perpendicular to its length produces high torque; applying force closer to the axis or parallel to the rod produces low or zero torque.

• Door Example: A student pushes at a point 0.7 m from the hinge with a 240 N force at 20° from the line to the door. The torque is:

$\tau = r F \sin(\theta) = (0.7)(240)\sin(70°) = 169\ \text{N·m}$

• Wrench Example: A wrench is used to tighten a nut, handle tilted 30° above horizontal, force of 100 N exerted straight down at 0.2 m from the axis:

$\tau = r F \sin(\theta) = (0.2)(100)\sin(60°) = 17.3\ \text{N·m}$

• Negative sign indicates rotation in the clockwise direction.

Net Torque and Rotational Equilibrium

Net Torque

If multiple forces act on an object, each can exert a torque about the pivot point. The net torque is the sum of the individual torques:

$\tau_{\text{net}} = \tau_1 + \tau_2 + \tau_3 + \ldots = \sum \tau$

• Net torque determines the angular acceleration of the object.

Rotational Equilibrium

An object is in rotational equilibrium if the net torque about any axis is zero:

$\sum \tau = 0$

• No rotation occurs if the sum of clockwise torques equals the sum of counterclockwise torques.

Example: Seesaw Problem

• A 150 lb man sits a distance $r$ from the center of a 12 ft long seesaw to balance a 40 lb child sitting at the other end.

• Choose the support as the axis of rotation.

• Each applied force is perpendicular to the radial line, so $\sin(90°) = 1$.

$\tau_{\text{CW}} = r (150) \sin(90°) = 150r$

$\tau_{\text{CCW}} = 6 (40) \sin(90°) = 240$

Set $\tau_{\text{CW}} = \tau_{\text{CCW}}$ for equilibrium:

$150r = 240$

$r = \frac{240}{150} = 1.6\ \text{ft}$

Summary Table: Key Torque Relationships

Quantity	Symbol	Formula	Units
Torque	$\tau$	$\tau = r F \sin(\theta)$	N·m
Net Torque	$\tau_{\text{net}}$	$\tau_{\text{net}} = \sum \tau$	N·m
Rotational Equilibrium		$\sum \tau = 0$	N·m

Additional info:

• Torque is a vector quantity; its direction is determined by the right-hand rule.

• Positive torque is typically defined as counterclockwise, negative as clockwise.

• Applications include levers, wrenches, doors, and seesaws.