Your niece from Problem 54 keeps her pet rabbit in a backyard hutch with thermal resistance \(0.25 \mathrm{K} / \mathrm{W}\). On a day when the outside temperature is \(-15^{\circ} \mathrm{C},\) she's worried that the rabbit's water will freeze, so you put a \(50-\) W heat lamp in the hutch. Will the bunny be able to drink its water? Neglect the heat due to the animal's metabolism.

Thermal resistance is a concept in heat transfer physics that quantifies the opposition to heat flow through a specific material. It's similar to electrical resistance, but instead of impeding the flow of electricity, it hinders the movement of heat. Imagine you're bundling up in layers on a cold day; each layer adds resistance against the chilly air, keeping you warmer.

The formula to express thermal resistance, denoted by the symbol 'R', is given by: \[ R = \frac{\Delta T}{P} \]where \(\Delta T\) is the temperature difference across the material, and 'P' is the power (or heat flow) through it.

In practical terms, the higher the thermal resistance, the better the material is at insulating, meaning it prevents heat from passing through it quickly. In the context of our exercise where the rabbit’s hutch has a thermal resistance of \(0.25 \mathrm{K} / \mathrm{W}\), it offers moderate insulation against the cold outdoor temperature, helping to keep the hutch warmer.

The principle of heat transfer is foundational to understanding how energy in the form of heat moves from one place to another. There are three primary modes of heat transfer: conduction, convection, and radiation. In our example, where a heat lamp is used, radiation is the main mode at play since the lamp emits heat waves that raise the temperature inside the rabbit's hutch.

Conduction would signify heat moving through the materials of the hutch's walls, while convection would involve the movement of warm air currents within the hutch. However, because we're focused on the heat lamp, we consider the heat energy radiating from the lamp to its surroundings.

The heat transfer principle applied in the step by step solution utilizes the relationship between thermal resistance (R), power output of the heat source (P), and the expected temperature change (\(\Delta T\)). This relationship is crucial in situations where you want to maintain or achieve a specific temperature, like ensuring a rabbit's water doesn't freeze on a cold day.

Temperature change calculation allows us to determine how much the temperature will increase or decrease in response to a heat source. The formula previously introduced, \(\Delta T = P \cdot R\), is a straightforward way to calculate the shift in temperature due to a heat source, considering the thermal resistance of the environment.

In the scenario with the rabbit's hutch, we were looking to calculate the resulting temperature inside the hutch after a 50-watt heat lamp was introduced, factoring in the thermal resistance of the hutch itself. By using the formula to find the temperature rise (\(\Delta T\)) and then adding this to the ambient temperature, we could confirm whether or not the hutch would stay above freezing and keep the rabbit's water liquid.