Question

A uniform square plate S(slide c) and a uniform rectangular plate R (sides b, a) have identical area and masses (Fig.) Show that

Answer 1

Given: m_R = m_S = m

Area of surface = Area of rectangular plate

or c² = \(a\times b=ab\)

I = mr²

From diagram, c>b, c²>b²

∴ \(\frac{b^2}{c^2}<1 \) or \((\frac{b}{c})^2<1 or I_{xR}<I_{xS}\)

(ii) \(\frac{I_{yR}}{I_{yS}}=\frac {m(\frac {a}{2})^2}{m(\frac {c}{2})^2}=\frac{a^2}{c^2}\)

as a > c, ∴ a² > c² or \((\frac{a}{c})^2>1\)

Hence, \(\frac{I_{yR}}{I_{yS}}>1\)

(iii) \(I_{zR}I_{xS} = m(\frac{d_R}{2})^2-m(\frac{d_S}{2})^2\)

= \(\frac{m}{4}[d^2_R-d^2_S]\)

= \(\frac{m}{4}\)(a² + b² - 2c²)

= \(\frac{m}{4}\)(a² + b² - 2ab)

[from c² = ab]

\(=\frac{(a-b)^2m}{4}>0\)

or \(\frac{I_{zR}}{I_{zS}}>0\,or \frac{I_{zR}}{I_{zS}}>1\)