To find the kinetic energy of the rotating body and compare it to the kinetic energy of a moving mass, we can start by calculating the kinetic energy of the rotating body using the formula for rotational kinetic energy, which is given by:
Rotational Kinetic Energy Formula
The formula for the rotational kinetic energy (KE_rot) is:
KE_rot = (1/2) * I * ω²
Where:
- I is the moment of inertia (in kg-m²)
- ω is the angular velocity (in rad/s)
Calculating the Rotational Kinetic Energy
In your scenario, the moment of inertia (I) is 3 kg-m² and the angular velocity (ω) is 3 rad/s. Plugging these values into the formula gives us:
KE_rot = (1/2) * 3 kg-m² * (3 rad/s)²
Now, let's compute it step by step:
- First, calculate (3 rad/s)² which equals 9 rad²/s².
- Next, multiply this by the moment of inertia: 3 kg-m² * 9 rad²/s² = 27 kg-m²·rad²/s².
- Finally, multiply by 1/2: (1/2) * 27 kg-m²·rad²/s² = 13.5 J.
Comparing to Translational Kinetic Energy
The next step is to compare this rotational kinetic energy to the kinetic energy of a moving mass, which is calculated using the formula:
KE_trans = (1/2) * m * v²
Where:
- m is the mass (in kg)
- v is the velocity (in m/s)
Finding the Matching Velocity
We know that the kinetic energy of the rotating body is 13.5 J, and we want to find a mass of 27 kg that has the same kinetic energy. Setting KE_trans equal to 13.5 J, we have:
13.5 J = (1/2) * 27 kg * v²
Now, let's solve for v²:
- Multiply both sides by 2: 27 kg * v² = 27 J.
- Now divide by 27 kg: v² = 1 J/kg.
- Taking the square root gives us v = 1 m/s.
Conclusion
Therefore, the speed of the moving mass that has the same kinetic energy as the rotating body is 1.0 m/s, which corresponds to option (a). This illustrates how rotational and translational kinetic energy can be interconnected through their respective formulas.
