Static Equilibrium: Bringing Balance to the Force (and Torque)

Credit: Public Domain

By Andrew Bennett

Early on in a first-year physics class, we learn that equilibrium is a state in which an object experiences no net force. As a result, that object does not accelerate.

What Is Rotational Equilibrium?

It turns out that this definition for equilibrium is not complete. It works well when the location of the force doesn't matter, which occurs when the object will not be rotating. Including the possibility for rotation adds another requirement for equilibrium. Instead of just having no linear acceleration, we must also have no angular (or rotational) acceleration. To achieve this, the net torque on an object must also be zero.

For example, if you placed a stick on the ground, then pushed on it equally with your hands in two opposite directions, the stick might not stay still. If your hands are at the same location on the stick, it will remain still. But if your hands are at different locations, the stick will begin to rotate when you push.

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How Are Torque and Gravity Related?

Early in a first-year physics class, we treat objects as though they don't take up any space. We don't consider where along the object forces are acting or any possibility for rotation. Once we begin to account for rotation, we need to rethink some of the forces we've dealt with before in terms of their location on the object and how this could contribute to the rotation of the object.

Gravity pulls downward on all objects near the Earth's surface. This force actually is a combination of many forces. Every object with mass attracts every other object with mass, down to the subatomic scale. We treat gravity on some objects as though it were a single force ... but where should that single force be placed?

Finding the Center of Gravity on an Object

Every object has a point where we say gravity is acting. We call this the center of gravity for the object (this is the same location as the center of mass). For simple geometric shapes, this will be in the geometric center of the object. More complex shapes, or shapes with varying mass densities, require calculus to find the center of mass. For now, we'll stick with the simpler case.

After finding the center of an object and placing the gravitational force at that location, we will often need to consider the contribution of gravity to the net torque on an object and how that affects the object's rotation.

Gravitational Force Example Problem

In this video, we define the center of gravity and practice adding a gravitational force vector to the free body diagrams for objects. Then, we calculate the torque caused by that force.

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What Is Static Equilibrium?

A typical problem in physics and engineering is to study an object in "static equilibrium."  This is a fancy way of saying that the object is still and will remain still. We can tease out that the object is not accelerating place to place (zero acceleration in the X and Y directions) and is not accelerating in its rotation (zero angular acceleration).

From that information, we can set up equations showing that the net force in the X-direction and in the Y-direction must be zero and that the net torque (for any chosen axis of rotation) must be zero.

Static Equilibrium Example Problem

In this video, we work through an example problem for an object in static equilibrium. The example is a classic problem that involves a beam resting on the top of a building and sticking out over the edge. A person begins to walk along the beam. The question is: How far out can they go before the beam starts to tip? I would not recommend trying this one out yourself!

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May the Force (and Torque) Be Zero

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