Question

using the law of conservation of energy,to prove that the total energy of a system is always conserved due to the change in its position?give few examples in favour of your anwer.

Answer 1

Task:

Use the law of conservation of energy to prove that the total energy of a system is always conserved due to the change in its position. Give few examples in favor of your answer.

Solution:

Example: The conservative system of two perfectly elastic bodies

Position 1

$$E _ {t o t a l} = \Sigma W _ {k i n e t i c} + \Sigma W _ {p o t e n t i a l} + \Sigma W _ {i n t e r n a l}$$ $$E _ {t o t a l} = \frac {m _ {1} v _ {1} ^ {2}}{2} + \frac {m _ {2} v _ {2} ^ {2}}{2} + m _ {2} g h + \Sigma W _ {i n t e r n a l}$$

Position 2

$$E _ {t o t a l} = \frac {m _ {1} v _ {1} ^ {\prime 2}}{2} + \frac {m _ {2} v _ {2} ^ {\prime 2}}{2} + m _ {1} g h + \Sigma W _ {i n t e r n a l}$$ $$h = \frac {v _ {2} ^ {\prime 2} - v _ {2} ^ {2}}{2 g} = \frac {v _ {1} ^ {2} - v _ {1} ^ {\prime 2}}{2 g}$$ $$v _ {2} ^ {\prime 2} = 2 g h + v _ {2} ^ {2}$$ $$v _ {1} ^ {2} - 2 g h = v _ {1} ^ {\prime 2}$$ $$\begin{array}{l} E _ {t o t a l} = \frac {m _ {1} (v _ {1} ^ {2} - 2 g h)}{2} + \frac {m _ {2} (2 g h + v _ {2} ^ {2})}{2} + m _ {1} g h + \Sigma W _ {i n t e r n a l} = \\ = \frac {m _ {1} v _ {1} ^ {2}}{2} + \frac {m _ {2} v _ {2} ^ {2}}{2} + m _ {2} g h + \Sigma W _ {i n t e r n a l} \\ \end{array}$$

The amount of energy remains the same