A 3.00-g copper coin at 25.0^∘ C drops 50.0 m to the ground....

A $3.00-\mathrm{g}$ copper coin at $25.0^{\circ} \mathrm{C}$ drops $50.0 \mathrm{m}$ to the ground. (a) Assuming $60.0 \%$ of the change in gravitational potential energy of the coin-Earth system goes into increasing the internal energy of the coin, determine the coin's final temperature. (b) What If? Does the result depend on the mass of the coin? Explain.

Key Concepts

Gravitational Potential Energy

This is the energy an object has because of its position in a gravitational field, typically calculated as the product of its mass, the gravitational acceleration, and its height (mgh). In problems where an object falls, its loss in gravitational potential energy is a key source of energy that can be transformed into other forms.

Energy Conversion Efficiency

This concept refers to the fraction of energy that is successfully converted from one form to another. In the context of the problem, it is the percentage of the gravitational potential energy change that becomes internal energy (increasing the coin's temperature), rather than being lost to the environment or other forms like sound or deformation.

Specific Heat Capacity

Specific heat capacity is a material property that indicates how much energy is required to raise the temperature of a unit mass of a substance by one degree. It is used to relate the energy added to a material to its change in temperature and appears in the equation Q = mc?T.

Mass Cancellation in Temperature Change

When calculating the temperature increase due to energy conversion using the equation ?T = Q/(mc), if the energy added (Q) is expressed in terms of mass (such as a fraction of mgh), the mass cancels out. This indicates that, under these conditions, the change in temperature is independent of the mass of the object.

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