Constraints and Reaction Forces

In the real world, objects are rarely free to move however they like. More often, they’re held in place or guided by other structures - like surfaces, ropes, or hinges. These limitations are called constraints, and they play a key role in how objects behave.

To understand how motion is controlled, we need to look at two essential ideas: the constraint itself, and the reaction force that comes with it.

What is a constraint?

A constraint is anything that limits how an object can move. It might block certain directions or force the object to move in a specific way.

Think of it as a rule the object has to follow. A surface might stop an object from falling. A string might keep it swinging in an arc. Constraints shape the way objects move by setting boundaries.

For example, a table supports an object and stops it from moving downward. A rope holds up a hanging object and prevents it from falling straight down.

What is a reaction force?

Whenever an object pushes against a constraint - like trying to fall through a surface - the constraint pushes back. That pushback is the reaction force.

It’s the force that stops the object from moving in a way it’s not allowed to. For solid surfaces, this is usually a force pointing directly away from the surface (called a normal force). In the case of a string or rod, it acts along the direction of the constraint.

Take a cup resting on a table. Gravity pulls the cup down, but the table pushes up with an equal and opposite force. That upward force is the normal reaction force $ N $, and it balances out the weight $ P $ of the cup.

So in short: the constraint is what limits motion, and the reaction force is how it enforces that limit. Understanding both helps us make sense of how objects stay in place, move, or resist motion in the world around us.