Q136 / 160 · AP EAMCET 2019 (22 Apr Shift 1) · Physics

Two metallic spheres \(P\) and \(Q\) are made of same material have same smoothness but the weight of \(P\) is 8 times that of \(Q\). If the two are heated to same temperature and left to cool, then the ratio of rate of cooling of \(Q\) to that of \(P\) is

  • A
    4
  • B
    8
  • Correct answer
    C
    2
  • D
    1

Solution

Given, weight of material \(P\) is 8 times the weight of material \(Q\). i.e., \(\quad m_p=8 m_Q\) From stefan's law, the rate of heat radiation from an object is given by \(\frac{\Delta Q}{\Delta t}=e \sigma A T^4\)...(i) where, \(e=\) emissivity, \(\sigma=\) Stefan's constant \(A=\text { area and } T=\text { Temperature }\) But, \(\Delta Q=m c \Delta T\)...(ii) \(\therefore\) From Eqs. (i) and (ii), we get \(\begin{aligned} \frac{m c \Delta T}{\Delta t} & =e \sigma A T^4 \\ \frac{\Delta T}{\Delta t} & =\frac{e \sigma A T^4}{m c} \quad \ldots (iii) \end{aligned}\) But, \(\quad m=\frac{4}{3} \pi r^3 \cdot \sigma\) \(\therefore \quad r=\left(\frac{3 m}{4 \pi \sigma}\right)^{\frac{1}{3}}\) \(\therefore \quad A=4 \pi r^2=4 \pi\left(\frac{3 m}{4 \pi \sigma}\right)^{\frac{2}{3}}\)...(iv) \(\therefore\) From Eqs. (iii) and (iv), we get \(\begin{aligned} & \frac{\Delta T}{\Delta t}=\frac{e \sigma T^4}{m c}\left[4 \pi\left(\frac{3 m}{4 \pi \sigma}\right)^{\frac{2}{3}}\right] \\ & \therefore \frac{\Delta T}{\Delta t}=k\left(\frac{1}{m}\right)^{\frac{1}{3}} \quad \text { where, } k=\left[\frac{e \sigma T^4}{c}\left(4 \pi\left(\frac{3}{4 \pi}\right)^{2 / 3}\right)\right] \\ & \therefore \frac{\left(\frac{\Delta T}{\Delta t}\right)_Q}{\left(\frac{\Delta T}{\Delta t}\right)_P}=\left(\frac{m_P}{m_Q}\right)^{\frac{1}{3}}=\left(\frac{8 m_Q}{m_Q}\right)^{\frac{1}{3}}=(8)^{\frac{1}{3}}=2 \end{aligned}\)

Hence, the correct answer is option: 'C'.

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