1. What is Rotational Kinetic Energy?
Just as a moving object has kinetic energy due to its linear motion, a rotating object has kinetic energy due to its rotation. This energy comes from the motion of all the particles inside the object as they rotate around an axis.
Every particle in a rotating body traces a circular path, and because each particle has its own speed, the total kinetic energy of the object is the sum of the kinetic energies of all those particles.
1.1. Everyday Examples
You observe rotational kinetic energy in many situations:
- A spinning wheel stores rotational energy.
- A flywheel in machines stores energy to keep the motion smooth.
- A rolling ball combines both translational and rotational kinetic energy.
- A windmill blade spinning in the wind has rotational kinetic energy.
1.1.1. Key Idea
The faster an object rotates or the farther its mass is from the axis, the more rotational kinetic energy it has.
2. Formula for Rotational Kinetic Energy
The rotational kinetic energy of a rigid body rotating with angular velocity \( \omega \) is given by:
\( K_r = \dfrac{1}{2} I \omega^2 \)
Where:
- \( I \) is the moment of inertia
- \( \omega \) is the angular velocity
2.1. Understanding the Formula
This expression shows that rotational kinetic energy depends on both how fast the object is spinning and how its mass is distributed. Objects with large moment of inertia store more energy at the same angular speed.
2.2. Comparison with Translational KE
Translational kinetic energy is given by \( K_t = \dfrac{1}{2} M v^2 \). Rotational kinetic energy has a similar form, but uses moment of inertia instead of mass, and angular velocity instead of linear speed.
3. Energy Distribution in Rolling Motion
When an object rolls without slipping, it possesses both translational and rotational kinetic energy. The total kinetic energy is:
\( K_{total} = \dfrac{1}{2} M v^2 + \dfrac{1}{2} I \omega^2 \)
This means rolling objects move slower than sliding objects because some energy goes into rotation.
3.1. Effect of Moment of Inertia
Objects with larger MI (like rings) rotate harder and therefore have more rotational kinetic energy, leaving less energy for translation. That is why they move slower down slopes.
3.2. Using No-Slip Condition
For rolling without slipping, \( v = \omega R \). Using this, rotational energy can be rewritten entirely in terms of linear velocity.
4. Rotational Kinetic Energy of Common Shapes
Different shapes store rotational energy differently because they have different moments of inertia.
4.1. Standard KE Expressions Using MI
- Solid Disc: \( K_r = \dfrac{1}{4} M R^2 \omega^2 \)
- Ring: \( K_r = \dfrac{1}{2} M R^2 \omega^2 \)
- Solid Sphere: \( K_r = \dfrac{1}{5} M R^2 \omega^2 \)
- Cylinder: \( K_r = \dfrac{1}{4} M R^2 \omega^2 \)
These expressions show how different shapes distribute their mass and how that affects rotational energy.
5. Relation Between Work, Torque and Rotational Energy
Just as force does work to change linear kinetic energy, torque does work to change rotational kinetic energy. The work done by torque is given by:
\( W = \tau \theta \)
This work increases the rotational kinetic energy of the body:
\( \tau \theta = \dfrac{1}{2} I \omega^2 - \dfrac{1}{2} I \omega_0^2 \)
5.1. Work–Energy Principle for Rotation
The rotational work–energy theorem states that the net work done by torque equals the change in rotational kinetic energy. This parallels the linear work–energy theorem.
6. Why Rotational Kinetic Energy Matters
Rotational kinetic energy plays a crucial role in understanding how rotating systems behave. It is important in machinery (flywheels, turbines), vehicles (wheels, engines), sports (spinning balls), and natural systems (planets and stars).
To fully understand rotational motion, we now turn to the topic of rotational dynamics, which combines torque, moment of inertia, and angular acceleration.