What is gyration length

The gyration length or radius of gyration about an axis is the radial distance from a point which would have the same moment of inertia as the body's actual distribution of mass if the body's total mass were concentrated at a point.

Explanation:

The gyration length appears to be the distance from a point where the whole body appears to be concentrated when it rotates about the point.

The gyration length can be illustrated this way.

Suppose we have a distribution of masses m₁, m₂, m₃,..., mₙ located at points r₁, r₂, r₃,..., rₙ respectively from a point O. Their moment of inertia I about point O is

I = m₁r₁² + m₂r₂² + m₃r₃² + ... + mₙrₙ²

If M = total mass = m₁ + m₂ + m₃ + ... + mₙ

Now I = MR² where R = gyration length

MR² = m₁r₁² + m₂r₂² + m₃r₃² + ... + mₙrₙ²

R² = m₁r₁² + m₂r₂² + m₃r₃² + ... + mₙrₙ²/M

R = √[(m₁r₁² + m₂r₂² + m₃r₃² + ... + mₙrₙ²)/(m₁ + m₂ + m₃ + ... + mₙ)]

R = √(∑mr²/∑m)

If the particles have the same mass, m₁ = m₂ = m₃ = ... = mₙ and M = nm. Since m = M/n

R = √[(mr₁² + mr₂² + mr₃² + ... + mrₙ²)/(m + m + m + ... + m)]

R = √[m(r₁² + r₂² + r₃² + ... + rₙ²)/nm]

R = √[(r₁² + r₂² + r₃² + ... + rₙ²)/n]

R = √(∑r²/n)

So the gyration length is the square-root of the sum of individual moment of inertias of the constituent masses divided by the sum of masses or the root mean square of the distances of the particles.