🌀 Rotational Kinetic Energy Calculator
Calculate rotational kinetic energy
How to Use This Calculator
Enter Moment of Inertia
Input the moment of inertia (I) in kg·m². This is the rotational equivalent of mass, measuring how mass is distributed relative to the rotation axis. For point masses, I = mr². For complex shapes, use the mass moment of inertia calculator first.
Enter Angular Velocity
Input the angular velocity (ω) in radians per second (rad/s). This is how fast the object is rotating. If you have RPM, convert using: ω = 2π × (RPM / 60). For example, 600 RPM = 62.83 rad/s.
Calculate and Review
Click the "Calculate" button to compute the rotational kinetic energy in joules (J). This is the energy associated with the object's rotation. Rotational kinetic energy is analogous to linear kinetic energy (½mv²) but for rotating objects.
Formula
KE_rotational = ½Iω²
Where:
• KE_rotational = Rotational kinetic energy (J)
• I = Moment of inertia (kg·m²)
• ω = Angular velocity (rad/s)
Compare to linear kinetic energy: KE_linear = ½mv²
Example 1: Spinning Disk
A disk with moment of inertia 2 kg·m² rotates at 20 rad/s. Calculate its rotational kinetic energy.
Given:
• Moment of inertia (I) = 2 kg·m²
• Angular velocity (ω) = 20 rad/s
Solution:
KE_rotational = ½Iω²
KE_rotational = ½ × 2 × 20²
KE_rotational = ½ × 2 × 400
KE_rotational = 400 J
Example 2: Rotating Wheel
A wheel with moment of inertia 0.5 kg·m² spins at 100 rad/s. What is its rotational kinetic energy?
Given:
• Moment of inertia (I) = 0.5 kg·m²
• Angular velocity (ω) = 100 rad/s
Solution:
KE_rotational = ½Iω²
KE_rotational = ½ × 0.5 × 100²
KE_rotational = ½ × 0.5 × 10,000
KE_rotational = 2,500 J
Frequently Asked Questions
What is rotational kinetic energy?
Rotational kinetic energy is the energy an object has due to its rotation. Just as linear kinetic energy (½mv²) describes the energy of translational motion, rotational kinetic energy (½Iω²) describes the energy of rotational motion. It depends on both the moment of inertia (how mass is distributed) and the angular velocity (how fast it spins).
What's the difference between rotational and linear kinetic energy?
Linear kinetic energy (KE = ½mv²) applies to objects moving in straight lines, while rotational kinetic energy (KE = ½Iω²) applies to rotating objects. An object can have both simultaneously - for example, a rolling wheel has both linear kinetic energy (center of mass moving) and rotational kinetic energy (spinning).
How does moment of inertia affect rotational kinetic energy?
Rotational kinetic energy is directly proportional to moment of inertia - doubling the moment of inertia doubles the energy (at the same angular velocity). The moment of inertia depends on how mass is distributed relative to the axis: mass farther from the axis contributes more to the moment of inertia.
Why is angular velocity squared in the formula?
The energy depends on the square of angular velocity because kinetic energy is always proportional to the square of velocity (whether linear or angular). This means doubling the angular velocity quadruples the rotational kinetic energy. This is why high-speed rotating objects can store significant amounts of energy.
Can an object have both rotational and linear kinetic energy?
Yes! Many objects have both. For example, a rolling ball has linear kinetic energy (its center moves forward) plus rotational kinetic energy (it spins). The total kinetic energy is the sum: KE_total = KE_linear + KE_rotational = ½mv² + ½Iω². This is why rolling objects move differently than sliding objects.
Where is rotational kinetic energy used in real-world applications?
Rotational kinetic energy is crucial in: flywheel energy storage systems, understanding the motion of rolling objects (wheels, balls), analyzing rotating machinery (turbines, engines), calculating the energy in spinning objects (gyroscopes, tops), understanding planetary motion and rotation, and designing energy-efficient rotating systems.
About Rotational Kinetic Energy Calculator
The rotational kinetic energy calculator computes the rotational kinetic energy using KE = ½Iω². Rotational kinetic energy is the energy associated with an object's rotation, analogous to linear kinetic energy for translational motion.
This calculator is essential for students studying rotational dynamics, engineers designing rotating systems, and anyone analyzing the energy of spinning objects. Understanding rotational kinetic energy helps explain everything from why rolling objects move differently than sliding ones to how flywheels store energy.