Moment of Inertia Calculation for Spheres Connected in a Square

How do we calculate the moment of inertia for spheres connected in a square?

Given the data of four spheres weighing 0.5 kg each connected in a square formation with sticks, how can we determine the moment of inertia for this system?

Answer:

The moment of inertia for spheres connected in a square is calculated depending on the position of the axis. If the axis passes centrally and perpendicularly, the inertia is 0.28 kg.m^2. If the axis divides the square in the middle, the inertia is 0.1 kg.m^2.

The moment of inertia, denoted by I, for any system of particles is calculated using the formula I = Σmr^2, where m is the mass of a particle and r is the distance of the particle from the axis of rotation. For this specific system of four spheres connected in a square, the moment of inertia can be calculated as follows: a-) When the axis passing through the center of the square and perpendicular to the square plane, the distance from each mass to the axis is half of the diagonal of the square. By using Pythagoras' theorem, the diagonal is calculated as √(0.4^2 + 0.4^2) = 0.56m. Therefore, the moment of inertia is 0.28 kg.m^2. b-) When the axis divides the square exactly in the middle, the distance from each mass to the axis is half of the side length of the square. Hence, the moment of inertia is calculated to be 0.1 kg.m^2.

Understanding the concept of moment of inertia helps in analyzing the rotational motion of objects and determining how their mass is distributed relative to the axis of rotation. It plays a crucial role in engineering, physics, and other fields where rotational dynamics are involved.