How heat is transmitted?
Heat is transmitted by conduction, convection, and radiation.
Perhaps you mean, How is heat transmitted by conduction. Atoms and molecules that are hot are more active, moving more rapidly than colder ones. They bump into nearby atoms & molecules causing them to move more rapidly and thus become hotter. This becomes a chain reaction so the heat is transmitted through the material.
What principle is being used when heat is transmitted between the boiler and cylinder in a hot water system?
All substances conduct heat to some degree as the heat is transmitted between atom via their vibration in solids. In electrically conductive materials such as metals and graphite, there are delocalised electrons which can pass from atom to atom freely. This is how charged is transmitted to mediate electricity. These flowing electrons can also carry heat and pass it more quickly from atom to atom than through just vibration alone.
Yes the vibration of molecules (heat) gets transmitted from one molecule to its neighbors and thus heat is conducted in all materials solids and liquids and gases alike. In the case of a solid the molecules that vibrate more (are hotter) cannot move as freely as in liquids and gases but they do bump into neighbors and share their energy (transfer heat).
Heat is transmitted in a gas less effectively because the atoms in a gas are spaced further apart, and are already (most of the time) heated up (the atoms are moving faster). A solid is tightly packed and it's atoms are slow moving. The faster the atom moves, the hotter it gets or is. This also contributes to the state it changes into (ie. solid, liquid or gas).
When sound hits a surface, it is either transmitted through the material or reflected back out, similar as light is. When sound is transmitted through matter, it is diffused (or spread out) by said matter; and also a small portion of the sound is converted to energy (such as heat). This is based on my personal understanding, it may not be 100% accurate but I you out.
Everything conducts heat, but a hot water bottle is designed to insulate, not conduct, so it doesn't lose heat too quickly and end up a cold water bottle. It does this by using water, which has a high heat capacity (a lot of energy must transmitted to cool it down), plastic, which is an insulator, and a furry coating, which traps still air, which prevents heat loss via convection.