Hot water is poured simultaneously into four metallic tumblers that are painted on the outside with rough white, rough black, shining white, and shining black paints. After some time, in which painted tumbler will the water be found to have cooled the most?

Introduction / Context:

This question tests basic ideas of heat transfer by radiation and the effect of colour and surface finish on cooling. When hot water is placed in metallic tumblers that are painted differently on the outside, the rate at which heat leaves the water through the tumbler surface depends strongly on whether the surface is dark or light, and rough or shiny. Understanding this concept is very important in everyday physics applications such as designing radiators, cooking utensils, and thermos flasks.

Given Data / Assumptions:

• Four metallic tumblers are identical in size, shape, and material.
• The outer surfaces are painted rough white, rough black, shining white, and shining black.
• The same quantity of hot water at the same initial temperature is poured simultaneously into each tumbler.
• Surrounding conditions such as room temperature and air movement are assumed to be the same for all four.

Concept / Approach:

Heat from the hot water is lost mainly through conduction through the metal and then radiation and convection from the outer painted surface. Since all tumblers are metallic and identical, the main difference in cooling rate comes from how effectively the outer paint allows radiant heat to be emitted. Black and rough surfaces are very good absorbers and emitters of thermal radiation. Shining and white surfaces are poor emitters and better reflectors of radiation. Therefore, the tumbler that behaves as the best radiator will cool the water fastest.

Step-by-Step Solution:

Verification / Alternative check:

A useful check is to recall practical examples. Radiators and the backs of some refrigerators are often painted dull black to enhance heat emission. On the other hand, thermos flasks and some cooking vessels use shiny or light coloured exteriors to reduce heat loss. These real life designs support the conclusion that a rough black surface promotes maximum cooling.

Why Other Options Are Wrong:

• Rough white: Roughness helps, but white colour is a poor emitter of radiation compared with black, so cooling is less than rough black.
• Shining white: Both white colour and shiny finish reduce emission, so this tumbler cools the water more slowly.
• Shining black: Black colour helps, but the shiny surface still reflects more radiation and emits less than a rough black surface.

Common Pitfalls:

Many learners think that shiny black surfaces must be the best emitter simply because they are black. However, shine reduces emissivity. Others focus only on colour and ignore the role of roughness versus smoothness. It is important to consider both factors together, which is why rough black is superior to shining black in terms of radiative heat loss.

Final Answer:

The water cools the most in the tumbler painted with a rough black surface.