Calculate the rotational inertia of a wheel that has a kinetic energy of 25, 000 J when rotating at 500 rev/min:’
The Correct Answer and Explanation is:
To calculate the rotational inertia (III) of the wheel, we can use the formula for the kinetic energy of rotational motion:KE=12Iω2KE = \frac{1}{2} I \omega^2KE=21Iω2
Where:
- KEKEKE is the kinetic energy of the wheel (25,000 J),
- III is the moment of inertia (which we need to find),
- ω\omegaω is the angular velocity in radians per second.
Step 1: Convert the rotational speed from revolutions per minute (rev/min) to radians per second.
The formula to convert from revolutions per minute (rev/min) to radians per second is:ω=2π×rev/min60\omega = \frac{2\pi \times \text{rev/min}}{60}ω=602π×rev/min
Given that the rotational speed is 500 rev/min:ω=2π×50060=1000π60≈52.36 rad/s\omega = \frac{2\pi \times 500}{60} = \frac{1000\pi}{60} \approx 52.36 \, \text{rad/s}ω=602π×500=601000π≈52.36rad/s
Step 2: Rearrange the kinetic energy equation to solve for the moment of inertia.
KE=12Iω2KE = \frac{1}{2} I \omega^2KE=21Iω2I=2KEω2I = \frac{2KE}{\omega^2}I=ω22KE
Substitute the values for KE=25,000KE = 25,000KE=25,000 J and ω=52.36\omega = 52.36ω=52.36 rad/s:I=2×25,000(52.36)2I = \frac{2 \times 25,000}{(52.36)^2}I=(52.36)22×25,000I=50,0002743.6≈18.22 kg⋅m2I = \frac{50,000}{2743.6} \approx 18.22 \, \text{kg} \cdot \text{m}^2I=2743.650,000≈18.22kg⋅m2
Final Answer:
The rotational inertia (III) of the wheel is approximately 18.22 kg·m².
Explanation:
The key to solving this problem lies in the relationship between kinetic energy and rotational motion. The formula KE=12Iω2KE = \frac{1}{2} I \omega^2KE=21Iω2 directly connects the kinetic energy to the moment of inertia and angular velocity. By first converting the rotational speed into radians per second, we can then isolate III in the equation and solve for it. The result gives the moment of inertia, which is a measure of the wheel’s resistance to changes in rotational motion, depending on both its mass and the distribution of that mass relative to the axis of rotation.
