A block is moving on an inclined plane making an angle $$45^\circ $$ with the horizontal and the coefficient of friction is $$\mu $$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define N = 10 $$\mu $$, then N is

Four solid spheres each of diameter $$\sqrt 5 $$ cm and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N $$ \times $$ 10⁻⁴ kg-m², then N is

5.6 liter of helium gas at STP is adiabatically compressed to 0.7 liter. Taking the initial temperature to be T₁, the work done in the process is

Steel wire of lenght ‘L’ at 40^oC is suspended from the ceiling and then a mass ‘m’ is hung from its free end. The wire is cooled down from 40^oC to 30^oC to regain its original length ‘L’. The coefficient of linear thermal expansion of the steel is 10⁻⁵ /^oC, Young’s modulus of steel is 10¹¹ N/m² and radius of the wire is 1 mm. Assume that L >> diameter of the wire. Then the value of ‘m’ in kg is nearly