Question

which is the correct energy equation including friction?
a ( w_{\text{ext}}=delta ke+delta pe_{g}+delta pe_{s}+delta t )
b ( w_{\text{ext}}=delta ke+delta pe_{g}+delta pe_{s}-w_{k} )
c ( w_{\text{ext}}+w_{k}=delta ke+delta pe_{g}+delta pe_{s} )
d ( w_{\text{ext}}-left|w_{k}
ight|=delta ke+delta pe_{g}+delta pe_{s} )
e all of the above

Explanation:

Step1: Analyze the work - energy principle with friction

The work - energy principle states that the net work done on a system is equal to the change in its mechanical energy (kinetic energy change \(\Delta KE\) and potential energy changes \(\Delta PE_g\) (gravitational) and \(\Delta PE_s\) (elastic)) plus the energy lost due to non - conservative forces (like friction). The work done by non - conservative forces (\(W_{nc}\)) can be written in different forms. If we consider the work done by external forces (\(W_{ext}\)) and the work done by friction (\(W_f\)) (where \(W_f\) is negative because it opposes motion), the general work - energy equation is \(W_{ext}+W_f=\Delta KE + \Delta PE_g+\Delta PE_s\).

Step2: Check each option

• Option A: \(W_{ext}=\Delta KE+\Delta PE_g+\Delta PE_s+\Delta T\). Here \(\Delta T\) can represent the thermal energy (energy lost due to friction). This is a valid form as \(W_f\) (work done by friction) is related to the generation of thermal energy.
• Option B: \(W_{ext}=\Delta KE+\Delta PE_g+\Delta PE_s - W_f\). Since \(W_f\) is negative (work done against friction), this is also a valid form (\(W_{ext}+W_f=\Delta KE+\Delta PE_g+\Delta PE_s\) can be rewritten as \(W_{ext}=\Delta KE+\Delta PE_g+\Delta PE_s+( - W_f)\) if we consider the magnitude of \(W_f\)).
• Option C: \(W_{ext}+W_f=\Delta KE+\Delta PE_g+\Delta PE_s\) is the standard work - energy equation with \(W_f\) (work done by friction, which is non - conservative).
• Option D: \(W_{ext}-\vert W_f\vert=\Delta KE+\Delta PE_g+\Delta PE_s\). Since \(W_f\) is negative (\(W_f =-\vert W_f\vert\)), this is equivalent to \(W_{ext}+W_f=\Delta KE+\Delta PE_g+\Delta PE_s\)

Answer:

E. All of the above