Moment of inertia and rotational kinetic energy

[CrazyNeutrino]

I can't seem to understand moment of inertia. What does it mean and how is it derived ?
How does it relate to rotational kinetic energy.

 

[tiny-tim]

Hi CrazyNeutrino!

From the PF Library on moment of inertia …

Moment of Inertia … relates rotational force (torque) to rotational acceleration in the same way that mass relates ordinary (linear) force to ordinary acceleration.​

ie τ = Iα just like F = ma

(and KE = 1/2 Iω² just like 1/2 mv²)

it is derived from I = ∫ dm r²

 

[CrazyNeutrino]

Why dm r^2?

 

[tiny-tim]

Why dm r^2?

because for a point mass m at distance r from the axis, subjected to a force F,

the torque is τ = Fr, and the angular acceleration is α = a/r

τ = Iα means Fr = Ia/r

but F = ma (good ol' Newton's second law)

so mar = Ia/r

so I = mr²​

(and moment of inertia of the whole = sum of moments of inertia of the parts, giving us ∫ dm r²)

 

[PJC01]

Moment of inertia is a measure of an object's resistance to rotational motion. It is derived from the distribution of mass within an object and is often compared to the concept of mass in linear motion. In linear motion, an object's mass determines its resistance to linear motion, while in rotational motion, an object's moment of inertia determines its resistance to rotational motion.

The moment of inertia is calculated by taking the sum of the products of each infinitesimal mass element within the object and its distance from the axis of rotation squared. This value is then multiplied by a constant depending on the shape of the object.

In simpler terms, moment of inertia can be thought of as the rotational equivalent of mass. Just as it takes more force to move a heavier object in linear motion, it takes more torque to rotate an object with a larger moment of inertia.

The relationship between moment of inertia and rotational kinetic energy is described by the rotational kinetic energy equation, which states that the rotational kinetic energy of an object is equal to half of its moment of inertia multiplied by the square of its angular velocity.

In other words, the larger the moment of inertia, the more rotational kinetic energy an object will have for a given angular velocity. This is because a larger moment of inertia means that the object has more resistance to rotation and therefore requires more energy to rotate at a certain speed.

I hope this explanation helps to clarify the concept of moment of inertia and its relationship to rotational kinetic energy. If you have any further questions, please feel free to ask.