The moment of inertia of a thin rod of mass M and length L, about an axis perpendicularto the rod at a distance $\frac{\mathrm{L}}{4}$from one end is

• $\frac{{\mathrm{ML}}^2}{6}$

• $\frac{{\mathrm{ML}}^2}{12}$

• $\frac{7 {\mathrm{ML}}^2}{24}$

• $\frac{7 {\mathrm{ML}}^2}{48}$

If the earth shrinks such that its mass does not change but radius decreases to one quarter of its original value then one complete day will take

• 96 h

• 48 h

• 6 h

• 1.5 h

A shell of mass 10 kg is moving with a velocity of 10 ms^-1 when it blasts and forms two parts of mass 9 kg and 1 kg respectively. If the 1^st mass is stationary, the velocity of the 2^nd is

• 1 m/s

• 10 m/s

• 100 m/s

• 1000 m/s

A circular disc rolls down on an inclined plane without slipping. What fraction of its total energy is translational ?

An infinite number of charges, each equal to q, are placed along the x-axis at x=1, x=2, x = 4, x=8 and so on. What is the potential at x = 0 due to this set of charges ?

An explosion breaks a rock into three parts in a horizontal plane. Two of them go off at right angles to each other. The first part of mass 1 kg moves with a speed of 12 ms^-1 and the second part of mass 2 kg moves with 8 ms^-1 speed. If the third part flies off with 4 ms^-1 speed, then its mass is

• 3 kg

• 5 kg

• 7 kg

• 17 kg

A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is

• $\frac{3\mathrm{g}}{2\mathrm{L}}$

• $\frac{\mathrm{g}}{\mathrm{L}}$

• $\frac{2\mathrm{g}}{\mathrm{L}}$

• $\frac{2\mathrm{g}}{3\mathrm{L}}$

A solid uniform sphere resting on a rough horizontal plane is given a horizontal impulse directed through its centre so that it starts sliding with an initial velocity v₀. When it finally starts rolling without slipping the speed of its centre is

• $\frac{2}{7} {\mathrm{v}}_0$

• $\frac{3}{7} {\mathrm{v}}_0$

• $\frac{5}{7} {\mathrm{v}}_0$

• $\frac{6}{7} {\mathrm{v}}_0$

259.

Three identical square plates rotate about the axes shown in the figure in such a way that their kinetic energies are equal. Each of the rotation axes passes through the centre of the square. Then the ratio of angular speeds ω₁ : ω₂ : ω₃ is

             

• 1 : 1 : 1

• $\sqrt{2} : \sqrt{2} : 1$

• $1 : \sqrt{2} : 1$

• $1 : 2 : \sqrt{2}$

A large number of particles are placed around the origin, each at a distance R from the origin. The distance of the centre of mass of the system from the origin is

• equal to R

• less than equal to R

• greater than R

• greater than equal to R