Answer to Question #259643 in Physics for Casper b

Question #259643

A metal sphere, when suspended in a constant temperature enclosure, cools from 80 0C to 70 0C in 5 minutes and cool from 70 0C to 62 0C in the next five minutes. Calculate the temperature of the enclosure.why did you divide the temperatures by two

1
Expert's answer
2021-11-01T12:53:14-0400

Below θ0\theta_0 is the temperature of the enclosure.

By Newton's law of cooling, during the cooling from 80 to 70°C, where θ1=80°C,θ2=70°C,dt=5 min\theta_1=80°\text C,\theta_2=70°\text C,\text dt=5\text{ min}:


dθdt=C(θθ0), θ1θ2dt=C(θ1+θ22θ0), 2=C(75θ0).\frac{\text d\theta}{\text dt}=C(\theta-\theta_0),\\\space\\ \frac{\theta_1-\theta_2}{\text dt}=C\bigg(\frac{\theta_1+\theta_2}{2}-\theta_0\bigg),\\\space\\ 2=C(75-\theta_0).



Cooling from 70 to 62°, where θ1=70°C,θ2=62°C,dt=5 min\theta_1=70°\text C,\theta_2=62°\text C,\text dt=5\text{ min}:


dθdt=C(θθ0), θ1θ2dt=C(θ1+θ22θ0), 1.6=C(66θ0).\frac{\text d\theta}{\text dt}=C(\theta-\theta_0),\\\space\\ \frac{\theta_1-\theta_2}{\text dt}=C\bigg(\frac{\theta_1+\theta_2}{2}-\theta_0\bigg),\\\space\\ 1.6=C(66-\theta_0).

Divide one equation by another:


21.6=C(75θ0)C(66θ0), θ0=30°C.\frac 2{1.6}=\frac{C(75-\theta_0)}{C(66-\theta_0)},\\\space\\ \theta_0=30°\text C.

In this problem, we put


θ=θ1+θ22\theta=\frac{\theta_1+\theta_2}{2}

because we need to use the average temperature of the system during the given interval.


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