Determine the moment of inertia for three teenagers weighing 60.0 kg, 45.0 kg, and 80.0 kg sitting at different points on the edge of a rotating stage, which has a radius of 12.0 m.
The moment of inertia of the system can be calculated by summing the individual moments of inertia of each person around the axis of rotation.
The moment of inertia of a point mass rotating around an axis at a distance r is given by:
I = mr^2
where I is the moment of inertia, m is the mass of the object, and r is the distance from the axis of rotation.
For the first teenager weighing 60.0 kg, the moment of inertia around the axis of rotation can be calculated as:
I1 = m1r1^2 = 60.0 kg x (12.0 m)^2 = 8,640 kg m^2
For the second teenager weighing 45.0 kg, the moment of inertia around the axis of rotation can be calculated as:
I2 = m2r2^2 = 45.0 kg x (12.0 m)^2 = 6,480 kg m^2
For the third teenager weighing 80.0 kg, the moment of inertia around the axis of rotation can be calculated as:
I3 = m3r3^2 = 80.0 kg x (12.0 m)^2 = 11,520 kg m^2
The total moment of inertia of the system can be found by summing the individual moments of inertia:
Itotal = I1 + I2 + I3 = 8,640 kg m^2 + 6,480 kg m^2 + 11,520 kg m^2 = 26,640 kg m^2
Therefore, the moment of inertia for three teenagers weighing 60.0 kg, 45.0 kg, and 80.0 kg sitting at different points on the edge of a rotating stage, which has a radius of 12.0 m, is 26,640 kg m^2.
To determine the moment of inertia for the teenagers sitting at different points on the edge of a rotating stage, we can use the parallel-axis theorem.
The moment of inertia for each teenager can be calculated as follows:
1. Teenager weighing 60.0 kg:
Moment of inertia = Mass * Distance²
Moment of inertia = 60.0 kg * (12.0 m)²
Moment of inertia = 60.0 kg * 144.0 m²
Moment of inertia = 8,640 kg m²
2. Teenager weighing 45.0 kg:
Moment of inertia = Mass * Distance²
Moment of inertia = 45.0 kg * (12.0 m)²
Moment of inertia = 45.0 kg * 144.0 m²
Moment of inertia = 6,480 kg m²
3. Teenager weighing 80.0 kg:
Moment of inertia = Mass * Distance²
Moment of inertia = 80.0 kg * (12.0 m)²
Moment of inertia = 80.0 kg * 144.0 m²
Moment of inertia = 11,520 kg m²
Therefore, the moment of inertia for the teenagers, weighing 60.0 kg, 45.0 kg, and 80.0 kg, sitting at different points on the edge of a rotating stage with a radius of 12.0 m, are approximately 8,640 kg m², 6,480 kg m², and 11,520 kg m², respectively.