What is the range of $r$ in the formula for the parallel axes theorem? Eg. if two discs are attached at a certain point on their circumference and both are in a plane, what is the moment of inertia of the system about the axis of one of the discs, so should I consider the other disc as a point mass located at $2R$ or apply parallel axes on it. In many such questions, they take the other body as a point mass and in some cases, they apply parallel axis theorem on it.
[Physics] When to take the other body as point mass or apply parallel axis theorem on it
moment of inertiareference framesrotational-dynamics
Best Answer
The Parallel Axis Theorem does not have a range. It applies regardless of how far an object is from the axis of rotation.
However, as the distance $r$ from the axis of rotation increases, treating the distant object as a point particle becomes a better and better approximation. Whether you decide to use the Parallel Axes Theorem or treat the distant object as a point particle depends on what level of accuracy you want to obtain.
Suppose the moment of inertia through the centre of mass is $I_0=Mk^2$ where $k$ is the radius of gyration, which is on the same order of magnitude as the width/breadth/length $d$ of the object. Then the moment of inertia about a parallel axis at distance $r$ is
$$I=I_0+Mr^2=M(k^2+r^2)$$ If $r \gg d \approx k$ then $I \approx Mr^2$ which is the moment of inertia of a point particle. The fractional level of accuracy you can expect from using the approximation is about $(\frac{d}{r})^2$. So if $r \approx 10d$ then the approximation will be accurate to about 1%.