find the radius of gyration of a uniform disc about an axis perpendicular to its plane and passing through its center.
The radius of gyration (k) of a uniform disc about an axis perpendicular to its plane and passing through its center is given by the formula:
k = (I / M)^0.5
where I is the moment of inertia of the disc and M is the mass of the disc.
The moment of inertia of a uniform disc about an axis perpendicular to its plane and passing through its center is given by the formula:
I = (1/2) * M * R^2
where M is the mass of the disc and R is the radius of the disc.
So, substituting the expression for I into the formula for k, we get:
k = (I / M)^0.5 = ((1/2) * M * R^2 / M)^0.5 = (R^2 / 2)^0.5 = R / (2^0.5) = R / 1.41
So, the radius of gyration of a uniform disc about an axis perpendicular to its plane and passing through its center is equal to R / 1.41, where R is the radius of the disc.