Radius of Gyration

The radius of gyration is an important topic to study rotational motion.

Suppose, a rigid body is rotating about an axis and I_1 is the moment of inertia of that body around the given axis. If we concentrate the whole rigid body mass to a point in such a way that the moment of inertia of the concentrated point (I_2)on the given axis becomes equal to the I_1, then the perpendicular distance between that point and the axis of rotation is the radius of gyration.

The radius of gyration is denoted by K.

For more clarity, here is a figure below:

radius of gyration explanation
fig: radius of gyration

We can define a radius of gyration of a rigid body about an axis as the perpendicular distance of a point where the whole mass of the body is supposed to be concentrated such that the moment of inertia of a point about the given axis becomes equal to the moment of inertia of the whole rigid body about the same axis.

Let I_1 be the moment of inertia of a rigid about the given axis then we can write I_1 as:

    \[I_1 = \sum_{i=1}^{n}m_ir_i^2\]

    \[I_1 = m_1r_1^2 + m_2r_2^2 + m_3r_3^2 + \ldots + m_nr_n^2\]

Now, suppose the whole mass of a rigid body is concentrated at a point at a perpendicular distance ‘K’ from the axis such that the moment of inertia of a point becomes equal to the above moment of inertia.

    \[I_2 = MK^2\]


    \begin{align*} I_2 &= I_1 \\ MK^2 &= m_1r_1^2 + m_2r_2^2 + m_3r_3^2 + \ldots + m_nr_n^2 \\ k^2 &= \frac{m_1r_1^2 + m_2r_2^2 + m_3r_3^2 + \ldots + m_nr_n^2}{M} \\ K &= \sqrt{\frac{m_1r_1^2 + m_2r_2^2 + m_3r_3^2 + \ldots + m_nr_n^2}{M}}\end{align*}

But for a uniform material rigid body, all the particles have equal masses. If 'n' is the number of particles and the masses m_1 = m_2 = m_3 = \ldots = m_n = m then,

    \begin{align*}K &= \sqrt{\frac{mr_1^2 + mr_2^2 + mr_3^2 + \ldots + mr_n^2}{nm}} \\ K &= \sqrt{\frac{m \left( r_1^2 + r_2^2 + r_3^2 + \ldots + r_n^2 \right)}{nm}} \\ \end{align*}

    \[ \boxed{K &= \sqrt{\frac{ r_1^2 + r_2^2 + r_3^2 + \ldots + r_n^2 }{n}}} \]

Hence, we can define radius gyration as the square root of the mean of the square of perpendicular distances of particles from the axis of rotation.

Points to remember on Radius of gyration

  • It is independent of the mass of a rigid body.
  • It only depends on the distribution of the masses.

Check the calculation of the moment of inertia as well as the radius of gyration of different rigid bodies from here.

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