How exactly does heat increase the kinetic energy of particles?
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The vibrations, rotations and general movement of particles will always contain some kinetic energy. We can measure this general vibration as a temperature of the particle system.
When you heating something, say water on a stove, the energy from the stove is being conducted through a vessel into the water, adding energy into it. The particles in the metal then vibrate faster and faster, and those metal particles in contact with the water will then impart energy to the water particles, heating the water.
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