Let's determine which equation correctly relates mechanical energy [tex]\( \text{ME} \)[/tex], thermal energy [tex]\( E_{\text{thermal}} \)[/tex], and total energy [tex]\( E_{\text{total}} \)[/tex] when friction is present in the system.
Understanding these energy forms:
1. Mechanical Energy (ME): This includes kinetic energy and potential energy.
2. Thermal Energy ([tex]\( E_{\text{thermal}} \)[/tex]): When friction is present, some of the mechanical energy is converted into thermal energy, increasing the heat or temperature of the system.
3. Total Energy ([tex]\( E_{\text{total}} \)[/tex]): This should account for all forms of energy in the system, including mechanical and thermal energy.
Let's review the options:
A. [tex]\( E_{\text{thermal }} = E_{\text{total }} + \text{ME} \)[/tex]
This equation suggests that thermal energy equals the total energy plus mechanical energy. Since total energy should already include both mechanical and thermal energy, adding mechanical energy again is incorrect. This equation does not correctly represent energy conservation.
B. [tex]\( E_{\text{thermal }} = \text{ME} - E_{\text{total }} \)[/tex]
This implies that thermal energy equals mechanical energy minus total energy. Again, this is not physically accurate because total energy should account for the sum of mechanical and thermal energy. This makes it impossible for the thermal energy to be a difference between mechanical and total energy.
C. [tex]\( E_{\text{thermal }} = E_{\text{total }} - \text{ME} \)[/tex]
This equation makes sense because it suggests that thermal energy is the total energy minus the mechanical energy. When friction is present, some mechanical energy is converted into thermal energy. Therefore, the mechanical energy decreases and is converted into thermal energy, keeping the total energy constant.
D. [tex]\( E_{\text{thermal }} = \frac{E_{\text{thermal }}}{\text{ME}} \)[/tex]
This equation implies a ratio where thermal energy equates to itself divided by mechanical energy, which is not relevant in the context of energy conversion due to friction.
Conclusion:
From the above analysis, the correct equation that relates mechanical energy, thermal energy, and total energy when friction is present is:
[tex]\[ E_{\text{thermal}} = E_{\text{total}} - \text{ME} \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{3 \text{ or C}} \][/tex]
