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 is K and the potential energy is U of an object then,
+ = +
where ; the subscripts i and f refer to the initial and final states of the system.
- Proof : When a conservative force does work W on an object within the system , that force transfers energy between kinetic energy K of the object and potential energy U of the system . From Work-Kinetic energy theorem , we can write ,
K = - = W
And from Work-Potential energy theorem , we can write,
U = - W
Now , combining this two equation , we find that ,
K = - U
We can rewrite this equation as ,
- = - ( - )
so, - = -
By rearranging this equation we find that ,
+ = +
- 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 ,
+ = +
or, m + mg= m + mg
Dividing by m and rearranging yield ,
= + 2g( - )
Putting = 0 and ( - ) = h leads to ,
= 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 = k .
- Experiments :