A thin metal disc of mass 0.5 kg and cross sectional area `0.05 m^2` is placed coaxially with a cylindrical block of height 0.5 m and same cross sectional area. The top face of the cylindrical block is maintained at a constant temperature of 300 K. If the initial temperature of the metal disc is 200 K, how long will it take for its temperature to raise to 250 K? The thermal conductivity of cylinder's material is `10 Wm^(-1) K^(-1)` and specific heat capacity of disc's metal is `600 J kg^(-1) K^(-1)`
Note: Consider that the thermal conductivity of disc is very high and the system is thermally insulated except the top face of cylinder.

Explanation

The given situation is shown below:

Here T is the temperature of the disc at any time t. The rate of heat flowing to the disc is
`(dQ)/(dt) = KA (T_0 - T) // h ` ...(i)
Due to flow of this heat, temperature of the disc increases by dT. Then,
`(dQ)/(dt) = ms ( (dT)/(dt) )` ...(ii)
Comparing (i) and (ii), we get
`KA (T_0 - T) // h = ms ((dT)/(dt) )`
` rArr (dT)/(dt) =(KA)/(hms) (T_0 -T)`
Integrating above equation, we get
` int_(T_1)^(T_2) (dT)/(T_0 - T) = (KA)/(hms_0) int_(0)^(t) dt`
where `T_1` and `T_2` are the initial and final temperatures of the metal disc, respectively, and t is the time taken
` rArr t = (hms)/(KA) ln ( (T_0 - T_1)/(T_0 - T_2) )`
Here , `T_1 = 200 K`
`T_2 = 250 K, T_0 = 300 K `
h = 0.5 m, m = 0.5 kg
`s = 600 J kg^(-1) K^(-1)`
`K = 10 W m^(-1) K^(-1)`
` A = 0.05 m^2`
` t = (0.5 xx 0.5 xx 600)/(10 xx 0.05) ln ( (300 - 200 )/(300 - 250) )`
` = 300 ln(2) = 208 s `