A planet ($P_1$) is moving around the star of mass $2M$ in the orbit of radius $R$. Another planet ($P_2$) is moving around another star of mass $4M$ in a orbit of radius $2R$. Ratio of time periods of revolution of $P_2$ and $P_1$ is ________.

A particle is rotating in a circular path and at any instant its motion can be described as

$\theta = \frac{5t^4}{40} - \frac{t^3}{3}$.

The angular acceleration of the particle after 10 seconds is _________ rad/s².

A parallel plate air capacitor has a capacitance $C$. When it is half filled as shown in figure with a dielectric constant $K = 5$, the percentage increase in the capacitance is ________.

Heat is supplied to a diatomic gas at constant pressure. Then the ratio of $\Delta Q : \Delta U : \Delta W$ is ______.