Question

Following the procedure used in Example 10.7 prove that the moment of inertia about the y axis of the rigid rod in Figure 10.15 is$\frac{1}{3}M{L}^2$.

Short answer

The moment of inertia of an uniform rigid rod $I=\frac{1}{3}M{L}^2$ is proved

Step-by-step solution

Moment of inertia.

The moment of inertia of an uniform rigid rod$I=\frac{1}{3}M{L}^2$is in its special form, when

The rotation axis is at one end of the rod$\left(h=0\right)$, then themoment of inertia of an uniform rigid rod $I=\frac{1}{3}M{L}^2$ .

Derivation of moment of inertia of an uniform rigid rod.

if the rod is cut into infinite number of parts, each having mass and length . So, moment of inertial, will be given as-

$dI=dm{x}^2$

From the formula of linear density, we have

$dm=\frac{M}{L}dx$

For the values of calculated above, we have-

$dI=\frac{M}{L}{x}^{2}dx$

Now,

$I=\int dI$

Substituting dI, (write the appropriate limits),

$I=\underset{-h}{\overset{L-h}{\int }}{x}^{2}dx$

After solving the integration,

$I=\frac{1}{3}M(L^2-3Lh-3h^2)$

Now, for the special cases,

When the rotation axis is at one end of the rod$\left(h=0\right)$, we have

$I=\frac{1}{3}M{L}^2$.

Therefore, the moment of inertia, along the y-axis at the edge of the rod is given as $I=\frac{1}{3}M{L}^2$