The moment of inertia (I) of a solid uniform disk is calculated using the formula: I = (1/2)mr². For a disk with a mass (m) of 5 kg and a radius (r) of 2.5 m, we can substitute these values into the formula.
Let’s perform the calculation:
- Mass, m = 5 kg
- Radius, r = 2.5 m
Now, substituting these into the moment of inertia formula:
I = (1/2) * 5 kg * (2.5 m)²
Calculating (2.5 m)² = 6.25 m², we have:
I = (1/2) * 5 kg * 6.25 m² = 15.625 kg·m²
This means the moment of inertia of the disk is 15.625 kg·m².
When the disk rolls without slipping down an inclined plane, this moment of inertia plays a crucial role in determining the angular acceleration and the dynamics of the motion. The relationship between linear and angular quantities in rolling motion is significant, and knowing the moment of inertia helps in understanding the energy distribution between translational and rotational motion.