A hot body placed in a surrounding of temperature θ0 obeys Newton's law of cooling (d θ)/(d t)=-

step 1 This question we have given a hot body which is placed in a surrounding of a temperature theta n

A hot body placed in a surrounding of temperature θ0 obeys...

A hot body placed in a surrounding of temperature $\theta_{0}$ obeys Newton's law of cooling $\frac{d \theta}{d t}=-k\left(\theta-\theta_{0}\right) .$ Its temperature at $t=0$ is $\theta_{1} .$ The specific heat capacity of the body is $s$ and its mass is $m$. Find (a) the maximum heat that the body can lose and (b) the time starting from $t=0$ in which it will lose $90 \%$ of this maximum heat.

Key Concepts

Heat Transfer and Energy Balance

Heat transfer involves the exchange of thermal energy between objects or between an object and its environment. In problems where a body cools, the total amount of heat lost can be calculated through the product of its mass, specific heat capacity, and the change in temperature, reflecting the energy balance involved in the cooling process.

Specific Heat Capacity

The specific heat capacity is a property of a material that indicates the amount of heat required to change the temperature of a unit mass of the substance by one degree. It is crucial for determining the total heat energy stored or lost by an object as its temperature changes.

Newton's Law of Cooling

Newton's Law of Cooling describes how the temperature of an object changes over time when placed in an environment at a different temperature. It postulates that the rate of change of the temperature difference between the object and its surroundings is proportional to that difference, leading to an exponential approach toward the ambient temperature.

Exponential Decay

Exponential decay refers to the process where a quantity decreases at a rate proportional to its current value. In the context of cooling, it is used to model how the temperature difference between the body and the environment reduces over time, following an exponential decay function.

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