Q: ABC is a plane lamina of the shape of an equilateral triangle . D, E are the mid points AB , AC and G is the centroid of the lamina . Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is I_{o} . If part ADE is removed the moment of inertia of the remaining part about the same axis is $\frac{N I_o}{16}$ where N is an integer . The value of N is …

Q: A uniform rod of length l is pivoted at one of its ends on a vertical shaft of negligible radius . When the shaft rotates at angular speed ω the rod makes an angle θ with it . To find θ equate the rate of change of angular momentum (direction going into paper) $\frac{m l^2}{12} \omega^2 sin\theta cos\theta $ about the center of mass (CM) to the torque provided by the horizontal and vertical forces F_{H} and F_{V} about the CM . The value of θ is then such that

(a) $\displaystyle cos\theta = \frac{2 g}{3 l \omega^2}$

(b) $\displaystyle cos\theta = \frac{g}{2 l \omega^2}$

Q: Two uniform circular discs are rotating independently in the same direction around their common axis passing through there centers . The moment of Inertia and angular velocity of the first disc are 0.1 kg m^{2} and 10 rad/s respectively while those for the second one are 0.2 kg m^{-2} and 5 rad/s respectively . At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed . he kinetic energy of the combined system is

Q: A uniformly thick wheel with moment on Inertia I and radius R is free to rotate about its center of mass . A massless string is wrapped over its rim and two blocks of masses m_{1} and m_{2} (m_{1} > m_{2}) are attached to the ends of the string . The system is released from rest . The angular speed of the wheel when m_{1} decends by a distance h is

Q: Three solid spheres each of mass m and diameter d are suck together such that the lines connecting the center form an equilateral triangle of side of length d . The ratio I_{o}/I_{A} of moment of Inertia I_{o} of the system about an axis passing the centroid and about center of any of the spheres I_{A} and perpendicular to the plane of the triangle is

(a) $\displaystyle \frac{15}{13}$

(b) $\displaystyle \frac{13}{15}$

(c) $\displaystyle \frac{13}{23}$

(d) $\displaystyle \frac{23}{13}$

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Ans: (c)

Sol: Let d is the diameter of each sphere and m be the mass of each sphere .

$\displaystyle CD = \sqrt{d^2 -\frac{d^2}{4}} = \frac{\sqrt{3}d}{2}$

$\displaystyle CO = \frac{2}{3} \times \frac{\sqrt{3}d}{2} = \frac{d}{\sqrt{3}}$

Moment of Inertia about O (using || axis theorem )