Two uniform solid spheres made of copper have radii 15 cm and 20 cm respectively. Both of them are heated to a temperature of `70^(@)C` and then placed in a region of ambient temperature equal to `45^(@)C` . What will be the ratio of the initial rates of cooling of both the spheres?

Answer

`01.33`

Explanation

For two bodies made up of same material, rate of loss of heat will depend only on their surface area for same temperature change.
`:. (((dQ)/(dt))_(1))/(((dQ)/(dt))_(2))= (A_(1))/(A_(2))=(r_(1)^(2))/(r_(2)^(2)).....(i)`
Ratio of initial rates of cooling= `(((dQ)/(dt))_(1))/(((dQ)/(dt))_(2))`
As `(dQ)/(dt)=mc (d theta)/(dt)`
`:. (m_(1)c_(1)(d theta//dt)_(1))/(m_(2)c_(2)(d theta//dt)_(2))=(r_(1)^(2))/(r_(2)^(2))`..... [ (from (i)]
since, both the spheres are made up of Copper,
`c_(1)=c_(2)=c`
`:. (((dQ)/(dt))_(1))/(((dQ)/(dt))_(2))=(r_(1)^(2)xxm_(2))/(r_(2)^(2)xxm_(1))`
As, spheres have same densities,
`(m_(1))/(m_(2))=(V_(1))/(V_(2))=(r_(1)^(3))/(r_(2)^(3))`
`:. ((d theta// dt)_(1))/((d theta//dt)_(2))=(r_(1)^(2))/(r_(2)^(2))xx(r_(2)^(3))/(r_(1)^(3))=(r_(2))/(r_(1))=(20)/(15)=(4)/(3)`
`1.33`