A wooden ball of density ρ is immersed in water of density σ to depth h and then released, then the height x above the surface of water upto which the ball jumps out of water is equal to

• $\left(\frac{\mathrm{\rho }}{\mathrm{\sigma }} - 1\right)\mathrm{h}$

• $\left(\frac{\mathrm{\sigma }}{\mathrm{\rho }} - 1\right)\mathrm{h}$

• $\left(1 - \frac{\mathrm{\sigma }}{\mathrm{\rho }}\right)\mathrm{h}$

• $\left(1 - \frac{\mathrm{\rho }}{\mathrm{\sigma }}\right)\mathrm{h}$

The dimensions of σb⁴ are (where, σ = Stefan's constant and b = Wein's constant).

• [ M⁰L⁰T⁰ ]

• [ ML⁴T^-3 ]

• [ ML^-2T ]

• [ ML⁶T^-3 ]

Two blocks of masses 2 kg and 5 kg are connected by an ideal string passing over a pulley. The block of mass 2 kg is free to slide on a surface inclined at an angle of 30° with the horizontal whereas 5 kg block hangs freely. The tension (T) in the string is

• 48.56 N

• 72.22 N

• 32.56 N

• 24.65 N

A sphere of mass m, moving with velocity v, enters in a hanging bag of sand and stops. If the mass of the bag is M and it is released by height h, then the velocity of the sphere was

• $\frac{\mathrm{M}}{\mathrm{m}}\sqrt{2\mathrm{gh}}$

• $\frac{\mathrm{m}}{\mathrm{M} + \mathrm{m}}\sqrt{2\mathrm{gh}}$

• $\frac{\mathrm{M} + \mathrm{m}}{\mathrm{m}}\sqrt{2\mathrm{gh}}$

• $\frac{\mathrm{m}}{\mathrm{M}}\sqrt{2\mathrm{gh}}$

A particle of mass m is projected with velocity v making an angle of 45° with the horizontal. When the particle lands on the level ground, the magnitude of the change in momentum is

• $\sqrt{2} \mathrm{mv}$along -ve y-direction

• $\sqrt{2} \mathrm{mv}$ along +ve y-direction

• $2\sqrt{2} \mathrm{mv}$ along -ve y-direction

• $2\sqrt{2} \mathrm{mv}$ along +ve y-direction

A car is speeding on a horizontal road curving round with a radius of 60 m. The coefficient of friction between the wheels and the road is 0.5. The height of centre of gravity of the car from the road level is 0.3 m and the distance between the wheels is 0.8 m. Will the vehicle skid or topple ?

• Yes, it will skid

• No, it will not skid

• Sometimes skid, sometimes not

• Data insufficient to calculate speed for toppling

The magnitude of vectors A, B and C are 3, 4 and 5 units respectively. If A + B =C, then the angle between A and B is

• $\frac{\mathrm{\pi }}{2}$

• cos^-1 0.6

• tan^-1 $\frac{7}{5}$

• $\frac{\mathrm{\pi }}{4}$

A ball is dropped to the ground from a height of 2 m. The coefficient of restitution is 0.6. To what height will the ball rebound ?

• 0.41 m

• 0.9 m

• 0.72 m

• 0.28 m

The gravitational field is given by E = − xi − 2y²j. When gravitational potential is zero at (0, 0), then the gravitational potential at (2, 1) is

• 3 unit

• 6 unit

• 9 unit

• 12 unit

The given graph shows the variation of velocity with displacement. Which one of the graphs given below correctly represents the variation of acceleration with displacement.