A diatomic ideal gas is compressed adiabatically $${1 \over {32}}$$ of its initial volume. If the initial temperature of the gas is T_i (in Kelvin) and the final temperature is a T_i, the value of $$a$$ is

A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 m long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is 50 N and the speed of sound is 320 ms⁻¹, the mass of the string is

A tiny spherical oil drop carrying a net charge $$q$$ is balanced in still air with a vertical uniform electric field of strength $${{81\pi } \over 7} \times {10^5}\,\,V{m^{ - 1}}.$$ When the field is switched off, the drop is observed to fall with terminal velocity $$2 \times {10^{ - 3}}\,\,m{s^{ - 1}}.$$ Given $$g = 9.8\,m\,{s^{ - 2}},$$ viscosity of the air $$ = 1.8 \times {10^{ - 5}}\,\,Ns\,{m^{ - 2}}$$ and the density of coil $$=900$$ $$kg$$ $${m^{ - 3}},$$ the magnitude of $$q$$ is

A uniformly charged thin spherical shell of radius $$R$$ carries uniform surface charge density of $$\sigma $$ per unit area. It is made of two hemispherical shells, held together by pressing them with force $$F$$ (see figure). $$F$$ is proportional to