10.4 Moment of Inertia and Rotational Kinetic Energy
3 min read•june 24, 2024
Rotational motion adds a twist to energy concepts. We'll spin through kinetic energy's two forms: translational and rotational. Both depend on mass and velocity, but rotational energy considers how mass is distributed around the .
is the star of the rotational show. It measures an object's resistance to , like a in an engine. The distribution of mass affects an object's , influencing its rotational behavior.
Rotational Motion and Energy
Rotational vs translational kinetic energy
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Translational kinetic energy KE=21mv2 depends on mass m and linear velocity v (car moving along a straight road)
KErot=21Iω2 depends on moment of inertia I and ω (spinning figure skater)
Both forms of kinetic energy measured in joules (J) and are scalar quantities
Total kinetic energy is the sum of translational and rotational kinetic energies (rolling wheel)
Moment of inertia fundamentals
Moment of inertia I measures an object's resistance to rotational acceleration (flywheel in an engine)
I=∑mr2 for discrete particles, where m is mass and r is distance from the axis of rotation
I=∫r2dm for continuous objects, where dm is an infinitesimal mass element
Objects with more mass farther from the axis of rotation have higher moment of inertia (figure skater with arms extended)
Higher moment of inertia results in greater resistance to rotational acceleration
Moment of inertia depends on the axis of rotation
Parallel-axis theorem I=[ICM](https://www.fiveableKeyTerm:ICM)+Md2 relates moment of inertia about any axis to the moment of inertia about the center of mass axis ICM, total mass M, and distance between axes d
Energy Conservation and Rotational Dynamics
Impact of inertia on rotational energy
Increasing moment of inertia while maintaining increases (figure skater pulling arms in during a spin)
Decreasing moment of inertia while maintaining angular velocity decreases rotational kinetic energy
If moment of inertia changes and rotational kinetic energy remains constant, angular velocity must change to compensate
Decreasing I increases ω, while increasing I decreases ω (figure skater extending arms to slow down spin)
L=Iω is conserved in the absence of external torques
Conservation in combined motion systems
Total mechanical energy [E = KE + KE_{rot} + PE](https://www.fiveableKeyTerm:E_=_KE_+_KE_{rot}_+_PE) includes translational kinetic energy KE, rotational kinetic energy KErot, and potential energy PE
In the absence of non-conservative forces, total mechanical energy is conserved [E_i = E_f](https://www.fiveableKeyTerm:E_i_=_E_f) (roller coaster at different heights)
Solve problems by equating initial and final total mechanical energy, considering changes in translational and rotational kinetic energy as well as potential energy
Account for energy transfers between translational, rotational, and potential forms (yo-yo rolling down a ramp)
The relates the work done on a system to its change in kinetic energy
Angular velocity in non-ideal rotations
Angular velocity ω=dtdθ is the rate of change of angular displacement θ with respect to time
In the presence of non-conservative forces, mechanical energy is not conserved
Work done by non-conservative forces, such as friction, reduces the total mechanical energy of the system (sliding block coming to rest due to friction)
To calculate angular velocity with non-conservative forces:
Calculate the work done by non-conservative forces [Wnc](https://www.fiveableKeyTerm:Wnc)
Subtract Wnc from the initial total mechanical energy to determine the final total mechanical energy
Use the final total mechanical energy to solve for the final angular velocity, considering the moment of inertia and any changes in potential energy (spinning top slowing down due to air resistance and friction at the point of contact)
Rotational Dynamics
τ=r×F is the rotational equivalent of force, causing rotational acceleration
Rotational acceleration α=dtdω is the rate of change of angular velocity
Newton's Second Law for rotation: τ=Iα, relating torque, moment of inertia, and rotational acceleration