a. The moment of inertia when the rod is rotated about its midpoint is 0.96 kg·m².
b. The moment of inertia when the rod is rotated at a point 0.25 m from the 4.4 kg mass is 20.04 kg·m².
c. The moment of inertia when the rod is rotated about the position of the 4.4 kg mass is 57.35 kg·m².
To compute the moment of inertia for different scenarios, we need to consider the distribution of masses and their distances from the axis of rotation. The moment of inertia (I) is calculated using the formula:
I = Σmr²
Where Σ represents the sum of individual contributions from each mass, m represents the mass, and r represents the perpendicular distance from the axis of rotation.
a. The rod is rotated about its midpoint.
- The masses are equidistant from the midpoint. Therefore, we divide the rod into two equal halves, with each half having a mass of 4.4 kg. The perpendicular distance from the midpoint to each mass is 0.98/2 = 0.49 m. Calculating the moment of inertia:
I = (4.4 kg * 0.49 m)² + (4.4 kg * 0.49 m)²
= 0.96 kg·m²
b. The rod is rotated at a point 0.25 m from the 4.4 kg mass.
- In this case, we consider the mass of 4.4 kg to be at the center and the mass of 7.7 kg to be at a distance of 0.98 - 0.25 = 0.73 m from the axis of rotation. Calculating the moment of inertia:
I = (4.4 kg * 0.25 m)² + (7.7 kg * 0.73 m)²
= 20.04 kg·m²
c. The rod is rotated about the position of the 4.4 kg mass.
- Here, we consider the 4.4 kg mass as the axis of rotation, and the 7.7 kg mass is at a distance of 0.98 m from the axis. Calculating the moment of inertia:
I = (7.7 kg * 0.98 m)²
= 57.35 kg·m²