### Related Formula According to Kepler's Third Law (Law of Periods): T^2 propto r^3$T^2 \propto r^3$ where: * T$T$ is the time period of revolution. * r$r$ is the orbital radius of the planet. ### Core Logic Using the proportionality relationship for two states: fracT_2^2T_1^2 = left( fracr_2r_1 right)^3$\frac{T_2^2}{T_1^2} = \left( \frac{r_2}{r_1} \right)^3$ Given: * T_1 = 200text days$T_1 = 200\text{ days}$ * r_2 = fracr_14$r_2 = \frac{r_1}{4}$ ### Step 1: Calculate the New Time Period Substitute the values into the proportionality relation: fracT_2^2(200)^2 = left( fracr_1 / 4r_1 right)^3 = left( frac14 right)^3 = frac164$\frac{T_2^2}{(200)^2} = \left( \frac{r_1 / 4}{r_1} \right)^3 = \left( \frac{1}{4} \right)^3 = \frac{1}{64}$ Taking the square root on both sides: fracT_2200 = sqrtfrac164 = frac18$\frac{T_2}{200} = \sqrt{\frac{1}{64}} = \frac{1}{8}$ T_2 = frac2008 = 25text days$T_2 = \frac{200}{8} = 25\text{ days}$ ### Pattern Recognition If orbital distance scales by x$x$, the period scales by x^3/2$x^{3/2}$. Here, distance scales by frac14$\frac{1}{4}$, so the period scales by left(frac14right)^3/2 = frac18$\left(\frac{1}{4}\right)^{3/2} = \frac{1}{8}$. Thus, 200 times frac18 = 25text days$200 \times \frac{1}{8} = 25\text{ days}$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Gravitation