(455) HL Rotational kinetic energy [IB Physics HL]

Rotational Kinetic Energy Overview

• Definition: Rotational kinetic energy is the energy of an object due to its rotation.

• Equations:

  • For rotational kinetic energy:

    • ( E_k = \frac{1}{2} I \Omega^2 )

    • Alternative form: ( E_k = \frac{L^2}{2I} )

  • Where:

    • ( E_k ) = rotational kinetic energy (joules)

    • ( I ) = moment of inertia (kg m²)

    • ( \Omega ) = angular velocity (radians/second)

    • ( L ) = angular momentum (kg m²/s)

Key Components

• Moment of Inertia: ( I = \frac{2}{5} m R^2 ) for a solid sphere (kg m²).

• Angular Momentum: ( L = I \Omega ) (kg m²/s).

• Units:

  • Kinetic energy: Joules

  • Moment of inertia: kg m²

  • Angular velocity: radians/second

  • Angular momentum: kg m²/s

Example Problem

• Situation: Two spheres, both with mass ( m ).

  • First sphere has radius ( R ) and angular velocity ( \Omega ). Discuss rotational kinetic energy ( E ).

  • Second sphere has radius ( 2R ) and angular velocity ( 3\Omega ). Find ( E_2 ) in terms of ( E ).

Old Sphere's Energy Calculation

• Initial equation:

  • ( E = \frac{1}{2} I \Omega^2 = \frac{1}{2} \left( \frac{2}{5} m R^2 \right) \Omega^2 )

  • Simplifies to:

  • ( E = \frac{1}{5} m R^2 \Omega^2 )

New Sphere's Energy Calculation

• New sphere:

  • ( E_2 = \frac{1}{2} I_2 (\omega_2)^2 )

  • Moment of inertia: ( I_2 = \frac{2}{5} m (2R)^2 = \frac{2}{5} m \cdot 4R^2 = \frac{8}{5} m R^2 )

  • New angular velocity: ( (3\Omega)^2 = 9\Omega^2 )

• Substitute in:

  • ( E_2 = \frac{1}{2} \left( \frac{8}{5} m R^2 \right) (9 \Omega^2) )

  • This simplifies to:

  • ( E_2 = \frac{36}{5} m R^2 \Omega^2 )

Ratio Calculation

• New over old:

  • ( \frac{E_2}{E} = \frac{\frac{36}{5} m R^2 \Omega^2}{\frac{1}{5} m R^2 \Omega^2} )

  • Cancels down to:

  • ( \frac{E_2}{E} = 36 )

• Conclusion:

  • ( E_2 = 36E )

Summary

• The rotational kinetic energy equation mirrors the linear kinetic energy equation, differing in variables: moment of inertia and angular velocity replace mass and velocity, respectively.

• Understanding the relationships and using ratios facilitates calculations in rotational dynamics.