For a mass $m$ attached to a spring with a spring constant $k$ we know that the period of simple harmonic oscillations will be:

$$
begin{equation}
T = 2 , pi ; sqrt{frac{m}{k}}
end{equation}
$$

We know that the mass is increased $m_2 > m_1$ and that the spring constant is decreased $k_2 < k_1$

Looking at formula (1) we see that:

$$
begin{align*}
T sim sqrt{m} \
T sim frac{1}{sqrt{k}}
end{align*}
$$

Since both the increase in mass and the decrease in the spring constant bring about an increase in the period we safely conclude that the period of the oscillation motion increases.

For a mass $m$ attached to a spring with a spring constant $k$ we know that the period of simple harmonic oscillations will be:

$$
begin{equation}
T = 2 , pi ; sqrt{frac{m}{k}}
end{equation}
$$

When a spring is compressed or pulled by a given force $F$ its elongation $x$ is proportional to that force, and the coefficient is called the spring constant.

This is described by Hooke’s Law:

$$
begin{equation}
F = k , x
end{equation}
$$

The mass attached to our spring $m = 0.213text{ kg}$

The number of oscillations $n = 102$

The time interval $t = 56.7text{ s}$

Firstly we find the period of the motion using the number of oscillations and time interval:

$$
begin{align*}
T = frac{t}{n} = frac{56.7text{ s}}{102} approx 0.556text{ s}
end{align*}
$$

Secondly, since we know the period and the mass we can find the spring constant of the spring using (1).

Rearranging for $k$ we get:

$$
begin{align*}
T = 2 , pi ; sqrt{frac{m}{k}} rightarrow k = left(frac{2 , pi}{T} right)^2 , m
end{align*}
$$

Plugging in the values we get:

$$
begin{align*}
k = frac{4 , pi^2}{(0.556text{ s})^2 } cdot 0.213text{ kg} = 27.2 ; frac{text{N}}{text{m}}
end{align*}
$$

Now to find the stretch distance $d$ we will use formula (2).

In our case the force $F$ stretching the spring is the weight of the attached mass $m$. Notice that the stretch distance $d$ is the same as the elongation $x$.

So we can write:

$$
begin{align*}
m , g = k , x
end{align*}
$$

Rearranging for $d$ we get:

$$
begin{align*}
d = frac{m , g}{k}
end{align*}
$$

Plugging in the values we get:

$$
begin{align*}
d &= frac{0.213text{ kg} cdot 9.81 ; frac{text{m}}{text{s}^2} }{27.2 ; frac{text{N}}{text{m}} } \
d &approx 0.077text{ m}
end{align*}
$$