A block of mass m is resting on a smooth horizontal surface. One end of a uniform rope of mass $\left(\frac{\mathrm{m}}{3}\right)$ is fixed to the block, which is pulled in the horizontal direction by applying force F at the other end. The tension in the middle of the rope is

• $\frac{8}{7} \mathrm{F}$

• $\frac{1}{7} \mathrm{F}$

• $\frac{7}{8} \mathrm{F}$

• $\frac{1}{5} \mathrm{F}$

A particle is acted upon by a force F which varies with position x as shown in figure. If the particle at x = 0 has kinetic energy of 25 J, then the kinetic energy of the particle at x =16 m is

• 45 J

• 30 J

• 70 J

• 135 J

Two springs P and Q of force constants k_p and k_Q $\left({\mathrm{k}}_{\mathrm{Q}} = \frac{{\mathrm{k}}_{\mathrm{p}}}{2}\right)$ are stretched by applying forces of equal magnitude. If the energy stored in Q is E, then the energy stored in P is

• E

• 2E

• $\frac{\mathrm{E}}{2}$

• $\frac{\mathrm{E}}{4}$

14.

A rod of mass m and length l is made to stand at an angle of 60° with the vertical. Potential energy of the rod in this position is

• mgl

• $\frac{\mathrm{mgl}}{2}$

• $\frac{\mathrm{mgl}}{3}$

• $\frac{\mathrm{mgl}}{4}$

From a circular ring of mass M and radius R, an arc corresponding to a 90° sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is k times MR². Then the value of k is

• $\frac{3}{4}$

• $\frac{7}{8}$

• $\frac{1}{4}$

• 1

A system consists of 3 particles each of mass m located at points (1, 1), (2, 2) and (3, 3). The coordinates of the centre of mass are

• (6,6)

• (3,3)

• (1,1)

• (2,2)

A wheel of moment of inertia 2.5 kg-m² has an initial angular velocity of 40 rads^-1. A constant torque of 10 Nm acts on the wheel. The time during which the wheel is accelerated to 60 rads^-1 is 

• 4 s

• 6 s

• 5 s

• 2.5 s

The ratio of radii of earth to another planet is $\frac{2}{3}$and the ratio of their mean densities is $\frac{4}{5}$. If an astronaut can jump to a maximum height of 1.5 m on the earth, with the same effort, the maximum height he can jump on the planet is

• 1 m

• 0.8 m

• 0.5 m

• 1.25 m

If an object of mass mis taken from the surface of earth (radius R) to a height 2R, then the work done is

• 2 mgR

• mgR

• $\frac{2}{3} \mathrm{mgR}$

• $\frac{3}{2} \mathrm{mgR}$

At what depth below the surface of the earth, the value of g is the same as that at a height of 5 km ?

• 1.25 km

• 2.5 km

• 10 km

• 7.5 km