In a typical combustion engine the workdone by a gas molecule is given by $$W = {\alpha ^2}\beta {e^{{{ - \beta {x^2}} \over {kT}}}}$$, where x is the displacement, k is the Boltzmann constant and T is the temperature. If $$\alpha$$ and $$\beta$$ are constants, dimensions of $$\alpha$$ will be :

If two similar springs each of spring constant K₁ are joined in series, the new spring constant and time period would be changed by a factor :

A particle is moving with uniform speed along the circumference of a circle of radius R under the action of a central fictitious force F which is inversely proportional to R³. Its time period of revolution will be given by :

The temperature $$\theta$$ at the junction of two insulating sheets, having thermal resistances R₁ and R₂ as well as top and bottom temperatures $$\theta$$₁ and $$\theta$$₂ (as shown in figure) is given by :