Show that the law of conservation of mechanical energy is obeyed by pulling or compressing the block tied at the end of a spring
(a)

Explanation

In figure (a) block is in equilibrium
` :. X = 0 `
When a block of mass m is pulled from its postion x = 0 upto `x_(m)` and it released then it moves from `-x_(m)` to `x_(m)`. The total mechanical energy at any point x = x it remains constant .
`1/2 kx_(m)^(2) = 1/2 kx^(2) +1/2 mv^(2)`
where v is velocity of block at x .
At original position of block at x = 0 , the speed is maximu and hence kinetic energy is maximum .
` :. 1/2mv_(m)^(2) =1/2 kx_(m)^(2)`
where `v_(m)` is maximum speed .
` :. v_(m)^(2) = k/m . x_(m)^(2)`
`:. v_(m) = sqrt(k/m) .x_(m)`
Dimensional formula of `sqrt(k/m) ` is `[T^(-2)]`
hence dimensional formula of `v_(m)` is same as the dimensional formula of `sqrt(k/m).x_(m)`
Hence , kinetic energy is converted into potential energy and potential energy is converted into kinetic energy but total mechanical energy remains constant .