Q4. If a body starts at a different temperature from its surroundings, does it ever reach exactly their temperature in a constant-K Newton model?
No. The excess temperature falls as C′e−Kt, which gets smaller and smaller but never reaches zero. In practice the body becomes too close to room temperature to tell the difference.
Q5. A body cools from 90 °C to 70 °C in 4 minutes in a room at 20 °C. Using the exact equation, what is its temperature after another 4 minutes?
In each 4 minutes the excess is multiplied by the same factor, 50/70. After the next 4 minutes the excess is 50 × (50/70), about 35.7 °C, so the temperature is about 55.7 °C.
Q6. For a body cooling with K = 0.05 per minute, how long does its excess temperature take to fall to half?
Half the excess means e−Kt = 1/2, so t = ln 2/K = 0.693/0.05, about 13.9 min, whatever the starting excess.
Q7. Two solid spheres of the same material, of radii r and 2r, are each nearly uniform in temperature and have the same temperature excess and heat-transfer coefficient per unit area. Which cools faster at first, and by what factor?
−dT2/dt = (k/ms)ΔT. Here k ∝ area ∝ r and m ∝ r, so the rate ∝ 1/r. The smaller sphere’s temperature falls 2 times as fast, even though the larger sphere loses 4 times as much heat per second.