A carnot engine takes $$3 \times {10^6}$$ cal. of heat from a reservoir at $${627^ \circ }C,$$ and gives it to a sink at $${27^ \circ }C$$. The work done by the engine is

According to Newton's law of cooling, the rate of cooling of a body is proportional to $${\left( {\Delta \theta } \right)^n},$$ where $${\Delta \theta }$$ is the difference of the temperature of the body and the surrounding, and $$n$$ is equal to :

The earth radiates in the infra-red region of the spectrum. The spectrum is correctly given by

A mass $$M$$ is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes $$SHM$$ of time period $$T.$$ If the mass is increased by $$m.$$ the time period becomes $${{5T} \over 3}$$. Then the ratio of $${{m} \over M}$$ is