Consider two uniform discs of the same thickness and different radii R1 = R and R2 = $$\alpha$$R made of the same material. If the ratio of their moments of inertia I1 and I2 , respectively, about their axes is I1 : I2 = 1 : 16 then the value of $$\alpha$$ is :

For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and O' (corner point) is :

A wheel is rotating freely with an angular speed $$\omega$$ on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is :

A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period T1 and, (ii) back and forth in a direction perpendicular to its plane, with a period T2. The ratio {{{T_1}} \over {{T_2}}} will be :

Four point masses, each of mass m, are fixed at the corners of a square of side $$l$$. The square is rotating with angular frequency $$\omega$$, about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is :

Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is M, radius of its top, R and height, H, then its moment of inertia about its axis is :

The linear mass density of a thin rod AB of length L varies from A to B as \lambda \left( x \right) = {\lambda _0}\left( {1 + {x \over L}} \right), where x is the distance from A. If M is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is :

Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is I = M\left( {{{{R^2}} \over 4} + {{{L^2}} \over {12}}} \right). If such a cylinder is to be made for a given mass of a material, the ratio {L \over R} for it to have minimum possible I is

The radius of gyration of a uniform rod of length $$l$$, about an axis passing through a point {l \over 4} away from the centre of the rod, and perpendicular to it, is :

As shown in the figure, a bob of mass m is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. When released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be:

Mass per unit area of a circular disc of radius $$a$$ depends on the distance r from its centre as $$\sigma \left(r\right)$$ = A + Br . The moment of inertia of the disc about the axis, perpendicular to the plane and assing through its centre is:

Consider a uniform rod of mass M = 4m and length $$\ell$$ pivoted about its centre. A mass m moving with velocity v making angle \theta = {\pi \over 4} to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is :

A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5.00 cm/s. Its kinetic energy is :

A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses m1 and m2 (m1 $$>$$ m2) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when m1 descents by a distance h is :

A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre ‘O’ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is :

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are 0.1 kg-m2 and 10 rad s–1 respectively while those for the second one are 0.2 kg-m2 and 5 rad s–1 respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is :

Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d. The ratio I0/IA of moment of inertia I0 of the system about an axis passing the centroid and about center of any of the spheres IA and perpendicular to the plane of the triangle is :

Identify the correct labels of A,B and C in the following graph from the options given below : Root mean square speed (Vrms); most probable speed (Vmp); Average speed (Vav)

Which one of the following graphs is not correct for ideal gas? d = Density, P = Pressure, T = Temperature

A mixture of one mole each of H2 , He and O2 each are enclosed in a cylinder of volume V at temperature T. If the partial pressure of H2 is 2 atm, the total pressure of the gases in the cylinder is :