In this question, we have to calculate the Force needed to stretch a spring. We know that the spring constant is 95 $dfrac{text{N}}{text{m}}$ and the length of the spring is 0.25 m.

Therefore, the formula to calculate the force needed to stretch the spring is as follow.

$mathbf{F = kx}$

where;

F = Force ,

k = spring constant and

x = length

1) Using the formula stated in cell 1.

$mathbf{F = kx}$

2) Substituting the vales in the formula:

F = (95) (0.25)

$$
boxed{ F = 24 text{N}}
$$

– Pendulum length: $l=1mathrm{~m}$

**Objective**
– Find the period $T$.

In order to find the period $T$, we need to relate it to the known information (length $l$ and acceleration due to gravity on Earth $a=g=9.81mathrm{~dfrac{m}{s^2}}$). To do this, we can use the formula:

$$begin{align}
T=2cdot picdotsqrt{dfrac{l}{g}}
end{align}$$

Using formula $(1)$, we can now simply calculate.

$$begin{aligned}
T&=2cdot picdotsqrt{dfrac{l}{g}}
\&=2cdot picdotsqrt{dfrac{1}{9.81}}
\&=boxed{2mathrm{~sec}}
end{aligned}$$

– Acceleration due to gravity: $g_{moon}=1.6mathrm{~dfrac{m}{s^2}}$
– Period: $T=2mathrm{~sec}$

**Objective**

– Find the length $L$.

In order to find length of pendulum $L$, we need to relate it to the given information ($g_{moon}$ and $T$). To do this, we can use the formula for period of the pendulum:

$$begin{align}
T=2cdot picdotsqrt{dfrac{L}{g}}
end{align}$$

Where $g$ is the acceleration due to gravity. We can now express the length $L$ from equation $(1)$ and calculate the result.

$$begin{aligned}
T&=2cdot picdotsqrt{dfrac{L}{g}}
\T^2&=4cdot pi^2cdot dfrac{L}{g}
end{aligned}$$

Now we can express $L$:
$$begin{align}
T^2&=4cdot pi^2cdot dfrac{L}{g}rightarrow L=dfrac{T^2cdot g}{4cdot pi^2}
end{align}$$

$$begin{aligned}
L&=dfrac{T^2cdot g_{moon}}{4cdot pi^2}
\&=dfrac{2^2cdot 1.6}{4cdot pi^2}
\&=boxed{0.16mathrm{~m}}
end{aligned}$$

– Pendulum length: $L=0.75mathrm{~m}$
– Period: $T=1.8mathrm{~sec}$

**Objective**

– Find the acceleration due to gravity $g$.

In order to find the acceleration due to gravity $g$, we need to relate it to the given information (Length $L$ and period $T$). To do this we can use the formula:

$$begin{align}
T=2cdot picdotsqrt{dfrac{L}{g}}
end{align}$$

In this formula, we know all the parameters except acceleration $g$, which we are looking for. So, we can just express $g$ from formula $(1)$, and calculate the result.

$$begin{align}
T^2=4cdotpi^2cdot{dfrac{L}{g}}
end{align}$$

Now we simply express the acceleration due to gravity $g$ from $(2)$:
$$begin{align}
g={dfrac{4cdotpi^2cdot L}{T^2}}
end{align}$$

Finally, using equation $(3)$, we can calculate acceleration due to gravity $g$.

$$begin{aligned}
g&={dfrac{4cdotpi^2cdot L}{T^2}}
\&={dfrac{4cdotpi^2cdot 0.75}{1.8^2}}
\&=boxed{9.14mathrm{~dfrac{m}{s^2}}}
end{aligned}$$

– First length of displacement: $l_1=0.40mathrm{~m}$
– Second length of displacement: $l_2=0.20mathrm{~m}$

**Objective**

– Find the change in potential energy.

To find the change in potential energy, we can first consider the formula for the potential energy of the spring $P$ due to displacement (stretch) of length $x$:

$$begin{align}
P=dfrac{1}{2}cdot kcdot x^2
end{align}$$
Where $k$ is the spring constant (different constant for different strings/materials).

$$begin{align}
P_1=dfrac{1}{2}cdot kcdot l_1^2
end{align}$$

$$begin{align}
P_2=dfrac{1}{2}cdot kcdot l_2^2
end{align}$$

The constant $k$ is the property of the spring, and since it is the same spring for both cases, it stays the same constant for $l_1$ and $l_2$.

$$begin{align}
dfrac{P_1}{P_2}&=dfrac{dfrac{1}{cancel 2}cdot cancel kcdot l_1^2} nonumber{dfrac{1}{cancel2}cdot cancel kcdot l_2^2}
\&=dfrac{l_1^2}{l_2^2}
end{align}$$

Using this result, we can simply calculate the proportion $dfrac{P_1}{P_2}$.

So, from this result, we see that $boxed{P_2=dfrac{ P_1}{4}}$, meaning that the potential energy stored at $0.2mathrm{~m}$ displacement is $4$ times smaller then energy stored at the $0,4mathrm{~m}$ displacement.