The law of the conservation of mechanical energy
Mechanical energy
The mechanical energy of a system is the sum of the potential energy and the kinetic energy of the system:
\[E_{\mathrm{M}} := E_{\mathrm{P}} + E_{\mathrm{K}}.\]
The law of conservation of mechanical energy
If only conservative forces act in an isolated system, the mechanical energy of the system remains constant.
If the system is not isolated or is under the effect of external forces, the mechanical energy of the system will change by an amount equal to the net external work done on the system:
\[\sum{W_{\mathrm{ext}}} = \Delta E_{\mathrm{M}}.\]
The work-kinetic energy principle is always valid, even if non-conservative forces, such as friction, are present in the system. When all forces are conservative and the system is isolated — that is, no external forces act on it — the net work \(\sum{W}\) done during changes within the system can be written as a change in potential energy. The advantage of this is that we only have to consider the initial and final positions, not the whole of the path.
Proof: the law of conservation of mechanical energy
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The net work done in the system as the state of the system changes from state (1) into state (2) is \(\sum{W} = W_{1 \rightarrow 2}.\) The potential energy in the initial state is \(E_{\mathrm{P}1} = W_{1 \rightarrow 0}\), whilst in the final state it is \(E_{\mathrm{P2}} = W_{2 \rightarrow 0}.\) The figure to the left shows the connexion between the net work done on the object and the change in the potential energy of the object. We can see that \(\sum{W} = E_{\mathrm{P}1} - E_{\mathrm{P}2}.\) The work-kinetic energy principle states that \[\sum{W} = \Delta E_{\mathrm{K}} = E_{\mathrm{K}2} - E_{\mathrm{K}1}.\] Substituting for \(\sum W\) into this formula yields \[E_{\mathrm{P1}} - E_{\mathrm{P2}} = E_{\mathrm{K2}} - E_{\mathrm{K1}}.\] Rearranging this, we get \[E_{\mathrm{P1}} + E_{\mathrm{K1}} = E_{\mathrm{P2}} + E_{\mathrm{K2}},\] that is, \[E_{\mathrm{M}1} = E_{\mathrm{M}2}.\] This is what we wanted to prove. |
