To solve the problem of correctly relating mechanical energy (ME), thermal energy ([tex]\(E_{\text{thermal}}\)[/tex]), and total energy ([tex]\(E_{\text{total}}\)[/tex]) when friction is present in the system, follow these steps:
1. Understand the Concepts:
- Mechanical Energy (ME): This is the sum of kinetic and potential energy in a system.
- Thermal Energy ([tex]\(E_{\text{thermal}}\)[/tex]): This energy results from friction converting mechanical energy into heat.
- Total Energy ([tex]\(E_{\text{total}}\)[/tex]): This is the sum of all types of energy within the system.
2. Relation Between Energies:
- When friction is present, mechanical energy is not completely conserved; some of it is transformed into thermal energy.
- The total energy in the system remains constant but its form can change. Specifically, the mechanical energy will decrease as some of it is converted to thermal energy.
3. Establish the Correct Equation:
- The system loses mechanical energy due to friction, and this lost energy manifests as thermal energy.
- Therefore, the correct relationship is that the thermal energy gained (caused by friction) is equal to the mechanical energy lost minus the total energy in the system.
4. Choose the Correct Option:
- We can rewrite the energy transformation process as:
[tex]\[
E_{\text{thermal}} = ME - E_{\text{total}}
\][/tex]
- This equation states that the thermal energy is the difference between the initial mechanical energy and the remaining total energy in the system once friction has done its work.
Given the options:
- A. [tex]\(E_{\text{thermal}} = E_{\text{total}} + ME\)[/tex]
- B. [tex]\(E_{\text{thermal}} = E_{\text{total}} - ME\)[/tex]
- C. [tex]\(E_{\text{thermal}} = ME - E_{\text{total}}\)[/tex]
- D. [tex]\(E_{\text{thermal}} = \frac{E_{\text{thermal}}}{ME}\)[/tex]
The correct answer is:
[tex]\[ \boxed{C} \quad E_{\text{thermal}} = ME - E_{\text{total}} \][/tex]
