9.2 The Second Condition for Equilibrium

Rotational Equilibrium and Torque

The first condition for equilibrium says the net force on an object must be zero, so it won't accelerate in any direction. But that's not enough. An object can have zero net force and still spin. The second condition for equilibrium addresses this: the net torque on the object must also be zero. Together, these two conditions guarantee that an object is in full static equilibrium, meaning it neither translates nor rotates.

Second Condition for Equilibrium

The second condition states that the sum of all torques acting on an object must equal zero:

$\sum \tau = 0$

While the first condition ($\sum F = 0$) prevents translational acceleration, the second condition prevents angular (rotational) acceleration. You need both to be satisfied for an object to be completely static.

Think about a beam supported at its center with a 10 N weight hanging 2 m to the left. The beam won't translate anywhere if a support force balances the weight, but it will rotate unless there's an equal torque on the right side. That's why the second condition matters: it's what keeps structures like bridges, cranes, and shelves from tipping or spinning.

Torque Calculation and Significance

Torque ($\tau$) is the rotational equivalent of force. Instead of pushing an object in a straight line, torque causes it to rotate about an axis. The formula is:

$\tau = rF\sin\theta$

• $r$ is the lever arm, the distance from the axis of rotation to the point where the force is applied
• $F$ is the magnitude of the applied force
• $\theta$ is the angle between the force vector and the lever arm

A few things to notice about this formula:

• Torque is maximized when the force is applied perpendicular to the lever arm ($\theta = 90°$, so $\sin\theta = 1$).
• If the force is applied directly along the lever arm ($\theta = 0°$), $\sin\theta = 0$ and the torque is zero. Pulling straight outward on a wrench handle won't turn the bolt.
• A longer lever arm produces more torque for the same force. This is why a longer wrench makes it easier to loosen a tight bolt.

Torque has a direction: counterclockwise is typically assigned as positive, and clockwise as negative. This sign convention lets you add torques algebraically.

Calculating Net Torque for Rotational Equilibrium

When you need to verify or use rotational equilibrium, follow these steps:

1. Choose an axis of rotation. You can pick any point, but choosing a point where an unknown force acts is a smart move because that force's torque becomes zero (its lever arm is zero), simplifying your math.
2. Identify all forces acting on the object and where they're applied.
3. Determine the lever arm ($r$) for each force, measured from the chosen axis to the force's point of application.
4. Calculate each torque using $\tau = rF\sin\theta$.
5. Assign signs. Counterclockwise torques are positive; clockwise torques are negative.
6. Set the sum equal to zero ($\sum \tau = 0$) and solve for the unknown.

Example: A 4.0 m uniform beam weighing 200 N is supported at its left end by a pivot. A cable pulls upward at the right end. The beam's weight acts at its center of gravity, 2.0 m from the pivot.

• Torque from weight: $\tau_w = (2.0\,\text{m})(200\,\text{N})\sin 90° = 400\,\text{N·m}$ (clockwise, so $-400\,\text{N·m}$)
• Torque from cable tension $T$: $\tau_T = (4.0\,\text{m})(T)\sin 90°$ (counterclockwise, so positive)
• Setting $\sum \tau = 0$: $4.0T - 400 = 0$, giving $T = 100\,\text{N}$

This same approach applies to see-saws, hanging signs, bridges, and any system where you need to find unknown forces that keep something from rotating.

Rotational Dynamics and Equilibrium

A few related concepts tie into this topic:

• Moment of inertia ($I$) describes how much an object resists changes in its rotation. It depends on both mass and how that mass is distributed relative to the axis.
• Newton's Second Law for rotation is $\sum \tau = I\alpha$, where $\alpha$ is angular acceleration. In static equilibrium, $\alpha = 0$, which is exactly why $\sum \tau = 0$.
• Angular momentum is conserved in any system with no net external torque. This isn't directly about statics, but it reinforces the idea that torque is what changes rotational motion.
• A rigid body in static equilibrium satisfies both conditions: no net force and no net torque. It experiences zero translational acceleration and zero angular acceleration.