J J Science Academy

Subject: Physics Topic: 1. Rotational Motion SubTopic: 1 1. M.I Of Discrete Particle System TestType: NEET

1. 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) $6ma^2$

$$I=[m \times 0^2]+[m \times a^2]+[m \times a^2]+[m \times(2a)^2]$$ $$=6ma^2$$

2. 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) $36kg-m^2$

Distance of masses from centroid $$x =\frac{2}{\sqrt{3}} m \\ I =2x^2+3x^2+4x^2=9x^2 \\ =9\left(\frac{2}{\sqrt{3}}\right)^2=9 \times\left(\frac{4}{3}\right)=12 kg-m^2$$

3. 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) $1.5kg-m^2$

Minimum M.I. $I_{CM}=\frac{m_1 m_2}{m_1+m_2} r^2$ $$=\frac{3(6)}{3+6}\left(\frac{1}{2}\right)^2$$ $$=\frac{18}{9} \times \frac{1}{4}=0.5 kg-m^2$$