At what temperature is the $$r.m.s$$ velocity of a hydrogen molecule equal to that of an oxygen molecule at $${47^ \circ }C?$$

Two spheres of the same material have radii $$1$$ $$m$$ and $$4$$ $$m$$ and temperatures $$4000$$ $$K$$ and $$2000$$ $$K$$ respectively. The ratio of the energy radiated per second by the first sphere to that by the second is

1 mole of a gas with $$\gamma = 7/5$$ is mixed with $$1$$ mole of a gas with $$\gamma = 5/3,$$ then the value of $$\gamma $$ for the resulting mixture is

If a spring has time period $$T,$$ and is cut into $$n$$ equal parts, then the time period of each part will be