Question

Moment of inertia of a sphere of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) is \(\mathrm{I}\). keeping mass constant if graph is plotted between \(\mathrm{I}\) and \(\mathrm{R}\) then its form would be.

Short answer

The graph between the moment of inertia (I) and radius (R) of a sphere while keeping the mass constant (M) would be a parabolic curve with the equation: \( I(R) = \dfrac{2}{5}M R^2 \)

Step-by-step solution

Moment of Inertia of a Solid Sphere

The moment of inertia of a given object depends on its mass distribution. For a solid sphere of mass M and radius R, the moment of inertia is given by the formula: \( I = \dfrac{2}{5}MR^2 \)

Study the relationship between I and R

To analyze the relationship between I (moment of Inertia) and R (radius) while keeping the mass M constant, we can rewrite the formula as: \( I(R) = \dfrac{2}{5}M R^2 \) Note that M is a constant value, so the graph of I with respect to R will be determined by the R^2 term.

Graphing I(R)

To plot I as a function of R, we see that I increases with R^2. This means that the graph will be a parabolic curve, with the vertex at the origin (0,0) and opening upwards (since R^2 is always positive). The graph will have the form of y = kx^2, where k is a constant, in this case, k = (2/5) * M. Therefore, the graph between the moment of inertia (I) and radius (R) of a sphere while keeping the mass constant M would be a parabolic curve with the equation: \( I(R) = \dfrac{2}{5}M R^2 \)