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| 1. |
Derivation of Newtons law of cooling. |
| Answer» Newton\'s Law of Cooling states that the temperature of a body changes at a rate proportional to the difference in temperature between its own temperature and the temperature of its surroundings. We can therefore writedTdt=−k(T−Ts)dTdt=−k(T−Ts)where,T = temperature of the body at any time, tTs = temperature of the surroundings (also called ambient temperature)To = initial temperature of the bodyk = constant of proportionality dTdt=−k(T−Ts)dTdt=−k(T−Ts)dTT−Ts=−kdtdTT−Ts=−kdtln(T−Ts)=−kt+lnCln\u2061(T−Ts)=−kt+ln\u2061Cln(T−Ts)=lne−kt+lnCln\u2061(T−Ts)=ln\u2061e−kt+ln\u2061Cln(T−To)=lnCe−ktln\u2061(T−To)=ln\u2061Ce−ktT−Ts=Ce−ktT−Ts=Ce−kt when t = 0, T = ToC=To−TsC=To−Ts Thus,T−Ts=(To−Ts)e−ktT−Ts=(To−Ts)e−ktT=Ts+(To−Ts)e−ktT=Ts+(To−Ts)e−ktThe formula above need not be memorized, it is more useful if you understand how we arrive to the formula. | |