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Chapter 20

Q. 20.8

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?

Step-by-Step

Verified Solution

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.

fig 20.21
fig 20.22
fig 20.23