In two-dimensional motion, which conditions are needed to be satisfied so the conservation of energy law holds? (for example, simple pendulum motion)
2 Answers
Mechanical energy is conserved when there are no non-conservative forces acting on the body. Examples are friction and elastic forces of stress in a body. These non-conservative forces convert mechanical energy to other forms of energy like heat and sound. So the mechanical energy is not conserved, while the total energy is.
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$\begingroup$ So, for simple pendulum, how can I show that total mechanical energy is conserved? I know that forces acting on a bob are gravitational force and the one on the string (I don't know what's the name of that force; maybe straining force, I'm not sure). $\endgroup$– govJul 29, 2013 at 16:16
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$\begingroup$ @Gorica, The force in the string is called tension force. Both gravity and tension are conservative forces. You should read about conservative forces on wikipedia $\endgroup$ Jul 29, 2013 at 16:19
Whenever the Lagrangian of the system doesn't explicitly depend on time; there is a conserved quantity which we call it energy.
$$\frac{dE}{dt}=0 \Leftrightarrow \frac{\partial L}{\partial t}=0$$
In this context, $E$ is defined as:
$$E = \sum_i p_i\dot{q_i}-L$$
Addendum: Neother's Theorem
Noether's Theorem is one of the most elegant theorems in theoretical physics, named after Emma Noether. It states that "If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time". In this case, conservation of energy can be found using this theorem, whenever we have symmetry in time.