Ncert Exercises & Solutions

Q. 27 A uniform square plate S (side c) and a uniform rectangular plate R
(sides b, a) have identical areas and masses.

Show that

$\begin{array}{lll} (a) I_{xR} / I_{xs} < 1 & (b) I_{yR} / I_{ys} > 1 & (c) I_{zR} / I_{25} > 1 \end{array}$

Hint: Moment of inertia, I $\propto$ mass.

Step 1: Find

\frac{I_{xA}}{I_{xS}} .

By given question

Area of square = Area of the rectangular plate

\Rightarrow c^{2} = a \times b \Rightarrow c^{2} = ab

Now by definition

(a)

\frac{I_{xA}}{I_{xS}} = \frac{b^{2}}{c^{2}} [∵ I \propto mass \propto (area)^{2}]

From the diagram b < c

\Rightarrow \frac{I_{xA}}{I_{xS}} = (\frac{b}{c})^{2} < 1 \Rightarrow I_{xβ} < I_{xS}

Step 2: Find

\frac{I_{y_{R}}}{I_{y_{S}}}

(b)

\begin{array}{l} \frac{I_{y_{R}}}{I_{y_{S}}} = \frac{a^{2}}{c^{2}} \\ as, a > c \\ \frac{I_{y_{R}}}{I_{y_{S}}} = (\frac{a}{c})^{2} > 1 \end{array}

Step 3: Find

I_{zR} - I_{zS} .

(c)

I_{zR} - I_{zS} \propto (a^{2} + b^{2} - 2 c^{2}) = a^{2} + b^{2} - 2 ab = (a - b)^{2} [∵ c^{2} = ab]

\Rightarrow (I_{zR} - I_{zS}) > 0 \Rightarrow \frac{I_{zR}}{I_{zS}} > 1