Question 20.8: Use the parallel axes theorem to find the moments of inertia...
Use the parallel axes theorem to find the moments of inertia of
i) a thin uniform rod of mass M and length h about a perpendicular axis through its end
ii) a thin uniform solid disc of mass M and radius r about
a) an axis perpendicular to its plane through a point on its circumference
b) a tangent.
iii) Which of the above is equally applicable to a solid cylinder?
i) For the rod I_{G} = \frac{1}{3}M\left(\frac{h}{2}\right)^{2} and the axes are a distance \frac{h}{2} apart. Hence
I_{A} = I_{G} + M\left(\frac{h}{2}\right)^{2}
= \frac{1}{3}M\left(\frac{h}{2}\right)^{2} + M\left(\frac{h}{2}\right)^{2}
= \frac{1}{3}Mh^{2}.
ii) a) In this case I_{G} = \frac{1}{2}Mr^{2} and the axes
are a distance r apart.
Hence I_{A} = I_{G} + Mr^{2}
= \frac{1}{2}Mr^{2} + Mr
= \frac{3}{2}Mr^{2}.
b) Now I_{G} = \frac{1}{4}Mr^{2}
⇒ I_{A} = \frac{1}{2}Mr^{2} + Mr
= \frac{5}{4}Mr^{2}.
iii) The result is applicable to a cylinder so long as this is formed by elongating the body in a direction parallel to the axis of rotation. It therefore applies in part ii) a) but not in the other cases.
