A uniform wooden stick of mass 1.6 kg and length $$l$$ rests in an inclined manner on a smooth, vertical wall of height h ( < $$l$$ ) such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of $$30^\circ $$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio $${h \over l}$$ and the frictional force f at the bottom of the stick are ( g =10 ms^-2 )

The position vector $$\overrightarrow r $$ of a particle of mass m is given by the following equation $$$\overrightarrow r \left( t \right) = \alpha {t^3}\widehat i + \beta {t^2}\widehat j,$$$where $$\alpha = {{10} \over 3}m{s^{ - 3}}$$, $$\beta = 5\,m{s^{ - 2}}$$ and m = 0.1 kg. At t = 1 s, which of the following statement(s) is(are) true about the particle?

Consider two solid spheres P and Q each of density 8 gm cm^–3 and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm^–3 and viscosity $$\eta $$ = 3 poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm^–3 and viscosity $$\eta $$ = 2 poiseulles. The ratio of the terminal velocities of P and Q is

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated (P) by the metal. The sensor has a scale that displays $${\log _2}\left( {{P \over {{P_0}}}} \right)$$, where P₀ is a constant. When the metal surface is at a temperature of 487^oC, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to 2767^oC?