To determine the correct equation that relates mechanical energy, thermal energy, and total energy in a system where friction is present, let's analyze each option step-by-step.
When friction is present, mechanical energy (ME) is partially converted into thermal energy (E_thermal). The total energy (E_total) in the system should account for both the mechanical energy and the thermal energy generated due to friction. Therefore, the total energy combines both forms of energy.
Let's go over each presented option:
Option A:
[tex]\[ E_{\text{total}} = ME - E_{\text{thermal}} \][/tex]
This option suggests subtracting the thermal energy from the mechanical energy to obtain the total energy. However, in the presence of friction, thermal energy is an addition to the system’s energy, not a subtraction.
Option B:
[tex]\[ E_{\text{total}} = ME \times E_{\text{thermal}} \][/tex]
This option suggests multiplying the mechanical energy by the thermal energy. Total energy is a sum of different energy forms and should not be a product of them.
Option C:
[tex]\[ E_{\text{total}} = E_{\text{thermal}} - ME \][/tex]
This option suggests subtracting the mechanical energy from the thermal energy to obtain the total energy, which does not correctly describe the conservation of energy in the system.
Option D:
[tex]\[ E_{\text{total}} = ME + E_{\text{thermal}} \][/tex]
This option suggests adding the mechanical energy to the thermal energy to get the total energy. When there is friction present, the mechanical energy is partially converted into thermal energy; thus, the sum of these energies gives the total energy in the system, which reflects the conservation of energy principle.
After analyzing each option, it is clear that:
[tex]\[ E_{\text{total}} = ME + E_{\text{thermal}} \][/tex]
Therefore, the correct equation that relates mechanical energy, thermal energy, and total energy in a system with friction is:
[tex]\[ \boxed{D} \][/tex]
This correctly represents the conservation of energy where the total energy accounts for both mechanical and the resulting thermal energy due to frictional forces.
