Does spring exert not force while in slack condition?

Question

Consider a block of mass $m$ hunged by a spring of spring constant $k$. Let it be in equilibrium position where $mg=kx$.

Now it is given a sudden impulse in upward direction so that it starts moving in upward direction, does the maximum height it would reach will be calculated as

$1/2mv^2=mgh$(where h is the max height reached)

Or the height reached will be different as spring will also exert force and thus involve itself in energy conservation equation. The above equation i think is only valid if there is no spring force throughout whole journey but this does not sit well with my intuition. So what is actually happening?

Answer 1

This problem is best addressed via energy. With $z$ as the height above the equilibrium position of the unloaded spring, the loaded system has potential:

$$ U(z) = \frac 1 2 kz^2 + mgz $$

The equilibrium position is:

$$ z_0 = -g\frac m k $$

so that:

$$ U(z_0)= \frac 1 2 kz_0^2 + mgz_0= -\frac 1 2 \frac{m^2g^2}{k}\equiv U_0$$

Now it's given an impulse. The details don't matter, all we need to know is the kinetic energy:

$$T_0= \frac 1 2 mv^2 $$

With that, you have the initial total energy:

$$ E_0 =U_0+T_0 = \frac 1 2 mv^2 - \frac 1 2 \frac{m^2g^2}{k} $$

$$E_0 =\frac m 2 \Big( v^2 - \frac{g^2}{\omega_0^2} \Big)$$

where $\omega_0=\sqrt{k/m}$.

As the mass moves:

$$ U(z)+\frac 1 2 m\dot z^2=E_0$$

which can solved for $z$ with $\dot z=0$.

Answer 2

At equilibrium, the spring force balances gravity, such that the behavior of the system is equivalent to that of mass $m$ and spring $k$ without gravity, just shifted by $x$. The maximum height (relative to equilibrium position) can be found by exchanging the initial kinetic energy with that absorbed by the spring. $g$ need not appear. In other words: $$\frac12mv^2 = \frac12kh^2$$

Or, keep gravitation potential energy in as well as spring energy; just take care that zero spring energy is placed at $x$ above equilibrium. Then, substitutions using the equilibrium statement will allow terms involving gravity to cancel each other out.