Why Isn't Work Included in the Sum of Forces in Dynamics?

[Marchese_alex]

One question about this excercise. Why "W" isn't included in the sum of forces? Is doing work because is in the same direction of the movement, so I would assume that both the force of spring and weight would be in in my sum of forces. But this is not the case here, why is that?

 

[NascentOxygen]

The weight of the body is partially opposed by the springs, but not fully. The resultant net force is downward and it accelerates the body downwards (but more slowly than if it were in free-fall). The force down is the weight, and the force up is that of two springs together with the inertial force (F = ma) of the body as it is accelerated.

You don't need to use calculus here if you memorize the formula that the energy stored in a linear spring is ½.kx² (and if you forget this formula you can work it out when needed like the author does here).