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.

the answer is kevlar

double the distance = divide strength by 4