Rotational Kinetic Energy
- Compute rotational kinetic energy with KE = ½Iω²
- Recognize a rolling object’s energy as translational plus rotational
- Include rotational energy in conservation-of-energy accounting
Spinning objects store kinetic energy
A rotating object has kinetic energy even if its center is not moving, because every bit of it is in motion. This rotational kinetic energy is KE = ½Iω² — the exact rotational twin of ½mv², with rotational inertia I in place of mass and angular speed ω in place of speed. A spinning flywheel can store large amounts of usable energy this way.
Rolling combines both kinds of motion
An object that rolls is doing two things at once: its center moves forward (translational KE = ½mv²) and it spins (rotational KE = ½Iω²). Its total kinetic energy is the sum of the two. That is why a rolling ball at the bottom of a ramp is moving slower than a frictionless sliding block from the same height — some of the energy went into spin.
A flywheel with rotational inertia 2 kg·m² spins at 3 rad/s. Find its rotational kinetic energy.
- 1.Use KE = ½Iω².
- 2.Square the angular speed: ω² = 3² = 9.
- 3.Substitute: KE = ½ × 2 × 9.
- 4.Compute: KE = 9 J.
When a round object rolls down a ramp, split its energy: mgh = ½mv² + ½Iω². Forgetting the rotational term makes you overestimate the final speed every time.
A flywheel with rotational inertia 4 kg·m² spins at 5 rad/s. What is its rotational kinetic energy?
A ball rolls without slipping down a hill. Its total kinetic energy at the bottom is best described as:
In a "race down the ramp" question, the object that puts less energy into rotation (smaller I relative to mR²) ends up moving faster. That is a direct consequence of splitting mgh between ½mv² and ½Iω².
Answer the 2 checkpoints as you read.
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