Paper Questions (Raw)
Q1. Four particles, each of mass m , are arranged in the (X-Y) plane. The moment of inertia of this array of masses, about Z axis, is
1) (ma^2)
2) (2ma^2)
3) (4ma^2)
4) (6ma^2)
Hint: $$I=[m \times 0^2]+[m \times a^2]+[m \times a^2]+[m \times(2a)^2]$$
$$=6ma^2$$
Q2. Masses (2kg), (3kg) and (4kg) are placed at the vertices of an equilateral triangle, of side (2m). The moment of inertia of the triangle about an axis through the centroid and perpendicular to the plane is
1) (6kg-m^2)
2) (8kg-m^2)
3) (12kg-m^2)
4) (36kg-m^2)
Hint: Distance of masses from centroid
$$x =\frac{2}{\sqrt{3}} m \\ I =2x^2+3x^2+4x^2=9x^2 \\ =9\left(\frac{2}{\sqrt{3}}\right)^2=9 \times\left(\frac{4}{3}\right)=12 kg-m^2$$
Q3. Two point masses, each of mass (3kg) and (6kg), are separated by a distance of (50cm). The minimum M.I. of system about an axis perpendicular to line joining them is
1) (0.15kg-m^2)
2) (0.5kg-m^2)
3) (1kg-m^2)
4) (1.5kg-m^2)
Hint: Minimum M.I.
(I_{CM}=\frac{m_1 m_2}{m_1+m_2} r^2) $$=\frac{3(6)}{3+6}\left(\frac{1}{2}\right)^2$$
$$=\frac{18}{9} \times \frac{1}{4}=0.5 kg-m^2$$