(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.