Now let us consider what form the work-energy theorem takes when both conservative and nonconservative forces act. We will see that the work done by nonconservative forces equals the change in the mechanical energy of a system. As noted in Kinetic Energy and the Work-Energy Theorem, the work-energy theorem states that the net work on a system equals the change in its kinetic energy, or Wnet=ΔKEWnet=ΔKE size 12{W rSub { size 8{"net"} } =D"KE"} {}. The net work is the sum of the work by nonconservative forces plus the work by conservative forces. That is,

Wnet=Wnc+Wc,Wnet=Wnc+Wc, size 12{W rSub { size 8{"net"} } =W rSub { size 8{"nc"} } +W rSub { size 8{c} } } {}

(1)so that

Wnc+Wc=ΔKE,Wnc+Wc=ΔKE, size 12{W rSub { size 8{"nc"} } +W rSub { size 8{c} } =Δ"KE"} {}

(2)where WncWnc size 12{W rSub { size 8{"nc"} } } {} is the total work done by all nonconservative forces and WcWc size 12{W rSub { size 8{c} } } {} is the total work done by all conservative forces.

Consider Figure 3, in which a person pushes a crate up a ramp and is opposed by friction. As in the previous section, we note that work done by a conservative force comes from a loss of gravitational potential energy, so that Wc=−ΔPEWc=−ΔPE size 12{W rSub { size 8{c} } = - Δ"PE"} {}. Substituting this equation into the previous one and solving for WncWnc size 12{W rSub { size 8{"nc"} } } {} gives

Wnc=ΔKE+ΔPE.Wnc=ΔKE+ΔPE. size 12{W rSub { size 8{"nc"} } =Δ"KE"+Δ"PE"} {}

(3)This equation means that the total mechanical energy (KE + PE)(KE + PE) size 12{ \( "KE + PE" \) } {} changes by exactly the amount of work done by nonconservative forces. In Figure 3, this is the work done by the person minus the work done by friction. So even if energy is not conserved for the system of interest (such as the crate), we know that an equal amount of work was done to cause the change in total mechanical energy.

We rearrange Wnc=ΔKE+ΔPEWnc=ΔKE+ΔPE size 12{W rSub { size 8{"nc"} } =D"KE"+D"PE"} {} to obtain

KEi+PEi+Wnc=
KE
f
+PEf.KEi+PEi+Wnc=
KE
f
+PEf. size 12{"KE""" lSub { size 8{i} } +"PE" rSub { size 8{i} } +W rSub { size 8{"nc"} } ="KE""" lSub { size 8{f} } +"PE" rSub { size 8{f} } } {}

(4)This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If WncWnc size 12{W rSub { size 8{"nc"} } } {} is positive, then mechanical energy is increased, such as when the person pushes the crate up the ramp in Figure 3. If WncWnc size 12{W rSub { size 8{"nc"} } } {} is negative, then mechanical energy is decreased, such as when the rock hits the ground in Figure 2(b). If WncWnc size 12{W rSub { size 8{"nc"} } } {} is zero, then mechanical energy is conserved, and nonconservative forces are balanced. For example, when you push a lawn mower at constant speed on level ground, your work done is removed by the work of friction, and the mower has a constant energy.

Comments:"This introductory, algebra-based, two-semester college physics book is grounded with real-world examples, illustrations, and explanations to help students grasp key, fundamental physics concepts. […]"