Maximum displacement in mass spring system

[Fitz Watson]

Homework Statement

A block is suspended by an ideal spring of the force constant K. If the block is pulled down by applying a constant force F and if maximum displacement of the block from its initial position of rest is X then, find the value of X.

Homework Equations

mg + F = XK + K(mg/K)

The Attempt at a Solution

Let mass of block = m
Before applying F, the block is at rest. Let extension of spring here be a. So,
aK = mg
Hence, a = (mg/K) ... {1}

Now, after the block comes in equilibrium under F,

F + mg = (X+a) K ..... {2}

Solving {1} & {2} gives:
X = F/KBut, my book says that X = 2F/K

 

[haruspex]

after the block comes in equilibrium under F,

It does not say that it comes to equilibrium. The force is maintained as a constant.

 

[Fitz Watson]

It does not say that it comes to equilibrium. The force is maintained as a constant.

But when it has reached maximum extension, doesn't it mean that now the spring force is equal to weight + external force applied. And that's why the block can't go further down

 

[haruspex]

But when it has reached maximum extension, doesn't it mean that now the spring force is equal to weight + external force applied.

Not necessarily. Think about KE.

 

[Fitz Watson]

Not necessarily. Think about KE.

Now that I rethink, will it be like after forces becoming equal, block will go further down due to inertia?

 

[haruspex]

Now that I rethink, will it be like after forces becoming equal, block will go further down due to inertia?

Yes.