# Conservation of Mechanical Energy

What is conservation of mechanical energy ?

- In an isolated system where only conservative forces cause changes, the kinetic energy and potential energy can change , but their sum , the mechanical energy
**E**of the system , cannot change .

Theory :If the kinetic energy isKand the potential energy isUof an object then,

\(K_{i}\) + \(U_{i}\) = \(K_{f}\) + \(U_{f}\)

where ; the subscriptsiandfrefer to the initial and final states of the system.

-Proof :When a conservative force does workWon an object within the system , that force transfers energy between kinetic energyKof the object and potential energyUof the system . FromWork-Kinetic energy theorem, we can write ,

\(\Delta\) K = \(K_{f}\) - \(K_{i}\) = W

And fromWork-Potential energy theorem, we can write,

\(\Delta\) U = - W

Now , combining this two equation , we find that ,

\(\Delta\) K = - \(\Delta\) U

We can rewrite this equation as ,

\(K_{f}\) - \(K_{i}\) = - ( \(U_{f}\) - \(U_{i}\) )

so, \(K_{f}\) - \(K_{i}\) = \(U_{i}\) - \(U_{f}\)

By rearranging this equation we find that ,

\(K_{i}\) + \(U_{i}\) = \(K_{f}\) + \(U_{f}\)

**Example :**A boy of mass 50 kg is released from rest at the top of a water slide , at height h=10 m above the bottom of the slide . Assuming that the slide is frictionless because of the water on it . Find the boy's speed at the bottom of the slide.**Answer :**We have ,

**\(K_{i}\) + \(U_{i}\) = \(K_{f}\) + \(U_{f}\)**

or, \(\frac{1}{2}\)m\(v_{f}^{2}\) + mg\(h_{f}\)= \(\frac{1}{2}\)m\(v_{i}^{2}\) + mg\(h_{i}\)

Dividing by m and rearranging yield ,

\(v_{f}^{2}\) = \(v_{i}^{2}\) + 2g( \(h_{i}\) - \(h_{f}\) )

Putting \(v_{i}\) = 0 and ( \(h_{i}\) - \(h_{f}\) ) = h leads to ,

\(v_{f}\) = \(\sqrt{2gh}\)

= \(\sqrt{2 \times 9.8 ms^{-2} \times 10 m}\)

= 14 m/s**Try it yourself :**A block of mass m = 2.0 kg is dropped from height h = 40 cm onto a spring of spring constant k= 1960 N/m . Find the maximum distance , the spring is compressed ? Note that , for a spring , U = \(\frac{1}{2}\)k\(x^{2}\) .**Experiments :**

**Cite as:**Conservation of Mechanical Energy.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/conservation-of-mechanical-energy/