Show that if a mechanical system consists of only one point, then its acceleration in an inertial coordinate system is equal to zero ("Newton's first law").

An inertial coordinate system is essential for understanding Newton's first law of motion. It is a frame of reference in which objects can either remain at rest or move with a consistent velocity when no external forces are applied. In simple terms, if you see a ball rolling smoothly across a floor without speeding up or slowing down, you can consider that it's likely in an inertial coordinate system.

Within this context, objects obey Newton's laws, making these systems ideal for analyzing mechanical phenomena. Imagine you're sitting in a car that's driving at a constant speed on a straight road; you feel no acceleration. That's because, relative to the car—an inertial reference frame—you're not experiencing any net force.

However, in a non-inertial frame, such as a car accelerating or decelerating, Newton's laws don't apply in their usual form, as objects would appear to have forces acting on them, even in the absence of external influences. This distinction is pivotal when trying to understand the behavior of objects under different observational frames.

A mechanical system consists of interacting parts that may include objects, constraints, and forces. The system you encounter in your textbook example is one of the simplest imaginable: it only includes a single point, with no size or shape, and no detailed structure to scrutinize. Such a simplified system is a stepping stone towards understanding more complex mechanics.

When we observe a mechanical system like this in an inertial coordinate system, we expect it to exhibit simple motion that aligns with Newton's laws. Real-world mechanical systems often comprise multiple parts—gears, levers, springs—and require intricate analysis. But grasping the concept with a single point lays the groundwork for future learning. When there's no external force, such a system in an inertial frame will not experience any acceleration.

Students studying mechanical systems learn to analyze the effects of forces and motion. But before one can deal with the complexities of, say, a car engine, understanding a singular, isolated point in an inertial frame is a fundamental first step. This simpler concept enables you to predict the behavior of more complex scenarios without getting wrapped up in complicated mathematical descriptions from the get-go.

Acceleration is a measure of how quickly an object's velocity changes. It can be an increase or decrease in speed, or a change in direction. When you press the gas pedal in your car, the acceleration pushes you back into your seat; this is a tangible example of acceleration at work.

In the context of your exercise, acceleration is particularly notable because its value is determined by the presence or absence of external forces. According to Newton's first law, if there's no net external force acting on an object in an inertial frame (like the point in your textbook example), it will not accelerate—meaning its acceleration must be zero.

The concept of acceleration is also connected to the inertia of an object—the property that makes it resist changes in its state of motion. This resistance is why a heavy object takes more force to accelerate than a lighter one. Remember, acceleration is not just about speed, but also direction. Even if something is moving at a constant speed, if it's changing direction, it's still accelerating, as is the case with an object moving in a circle at constant speed.