The centre of mass of three particles of mass m₁ = 1.0 kg, m₂ = 2.0 kg, and m₃ = 3.0 kg at the corners of an equilateral triangle 1.0 m on a side, as shown in figure

• $\left(\frac{7}{12}\mathrm{m}, \frac{\sqrt{3}}{4}\mathrm{m}\right)$

• $\left(\frac{5}{12}\mathrm{m}, \frac{\sqrt{3}}{5}\mathrm{m}\right)$

• $\left(\frac{5}{13}\mathrm{m}, \frac{\sqrt{3}}{5}\mathrm{m}\right)$

• $\left(\frac{8}{12}\mathrm{m}, \frac{\sqrt{3}}{5}\mathrm{m}\right)$

A system consists of two point masses M and m (< M). The centre of mass of the system is

• at the middle of m and M

• nearer to M

• nearer to m

• at the position of large mass

Two perfectly elastic particles A and B of equal masses travelling along the line joining, them with velocity 15 m/s and 20 m/s respectively collide. Their velocities after the elastic collision will be (in m/s) respectively.

• 0 and 25

• 5 and 20

• 10 and 15

• 20 and 15

If the equation for the displacement of a particle moving on a circular path is given by θ = 2 t³+ 0.5, where θ is in radian and t in second, then the angular velocity of the particle after 2 s from its start is

• 8 rad/s

• 12 rad/s

• 24 rad/s

• 36 rad/s

The moment of inertia of a uniform ring of mass m and radius r about an axis, AA' touching the ring tangentially and lying in the plane of the ring only, as shown in figure, is

• $\frac{5}{4} {\mathrm{mr}}^2$

• $\frac{5}{2} {\mathrm{mr}}^2$

• $\frac{1}{4} {\mathrm{mr}}^2$

• $\frac{1}{2} {\mathrm{mr}}^2$

If the disc shown in figure has mass M and it is free to rotate about its symmetrical axis passing through O, its angular acceleration is

• $\frac{6\mathrm{F}}{\mathrm{MR}}$

• $\frac{3\mathrm{F}}{\mathrm{MR}}$

• $\frac{6\mathrm{F}}{5\mathrm{MR}}$

• $\frac{4\mathrm{F}}{\mathrm{MR}}$

A flywheel of moment of inertia 0.4 kg-m² and radius 20 cm is free to rotate about a central axis. If a string is wrapped on its circumference and it is pulled with a force of 10 N, then the change in its angular velocity after 4 s will be

• 10 rad/s

• 20 rad/s

• 40 rad/s

• 5 rad/s

Three identical spheres of mass m each are placed at the corners of an equilateral triangle of side 1 m. Taking one of the corners as the origin, the position vector of centre of mass is

• $\frac{1}{2}\stackrel{\wedge }{\mathrm{i}} + \frac{\sqrt{3}}{2} \stackrel{\wedge }{\mathrm{j}}$

• $\frac{1}{2} \stackrel{\wedge }{\mathrm{i}} + \frac{1}{2\sqrt{3}} \stackrel{\wedge }{\mathrm{j}}$

• $\stackrel{\wedge }{\mathrm{i}} + \stackrel{\wedge }{\mathrm{j}}$

• $2\stackrel{\wedge }{\mathrm{i}} - \stackrel{\wedge }{\mathrm{j}}$

A disc revolves with a speed of $33\frac{1}{3}$ rev min^-1 and a radius of 15 cm. Two coins are placed at 4 cm and 14 cm away from the centre of the disc. If coefficient of friction between the coins and the disc is 0.15, then which of the two coins will revolve with the disc?

• coin placed at a distance 4 cm

• coin placed at a distance 14 cm

• both coins

• Neither one of the coins

A uniform sphere of mass 500 g rolls down a plane surface without slipping on it, so that its centre moves at a speed of 0.02ms^-1 . The total kinetic energy of rolling sphere would be (in J)

• 1.4 × 10^-4 J

• 0.75 × 10^-3 J

• 5.75 × 10^-3 J

• 4.9 × 10^-5 J