A black coloured solid sphere of radius R and mass M is inside a...

\dfrac{d\theta}{dt} = - \dfrac{\sigmaAe}{Ms}\left({\theta}^{4} - {\theta}_{0}^{4}\right)$<br /><br /><img alt="" src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/445148556545.png" style="width: 200px; height: 400px;" /><br /><br />$\dfrac{d\theta}{dt} = - \dfrac{\sigmaAe}{M\alpha{\theta}^{3}}\left({\theta}^{4} - {\theta}_{0}^{4}\right)$</p><p>$∫ \dfrac{{\theta}^{3}}{\left({\theta}^{4} - {\theta}_{0}^{4}\right)}d\theta = - \dfrac{\sigmaAe}{M\alpha} ∫ dt$</p><p>Let \left({\theta}^{4} - {\theta}_{0}^{4}\right) = x$

$4{\theta}^{3}d\theta = dx$

$∫ \dfrac{dx}{4x} = - \dfrac{\sigmaAe}{M\alpha}{\left[t\right]}_{0}^{t}$

$\dfrac{1}{4}\left[\ln x\right] = \dfrac{M\sigmaAe}{M\alpha}{\left[t\right]}_{0}^{t}$

$\dfrac{1}{4}\left[\ln \left({\theta}^{4} - {\theta}_{0}^{4}\right)\right]$$\begin{array}{ll}2{T}_{0} \\ 3{T}_{0}\end{array}$$= - \dfrac{\sigmaAe}{M\alpha}t$

$\dfrac{1}{4}\ln \left(\dfrac{{\left(2{T}_{0}\right)}^{4} - {T}_{0}^{4}}{{\left(3{T}_{0}\right)}^{4} - {T}_{0}^{4}}\right) = - \dfrac{\sigmaAe}{M\alpha}t$

$t = - \dfrac{\alphaM}{4\sigmaAe}\ln \left(\dfrac{15{T}_{0}^{4}}{80{T}_{0}^{4}}\right)$

$= - \dfrac{\alphaM}{4\sigmaAe}\ln \left(\dfrac{3}{16}\right)$

$= - \dfrac{\alphaM}{4\sigma\left(4\pi{R}^{2}\right)e}\ln \left(\dfrac{3}{16}\right)$ where $e = 1$

$t = \dfrac{\alphaM}{16\pi\sigma{R}^{2}}\ln \left(\dfrac{16}{3}\right)$