====== 2. Gravitational acceleration from a pendulum ======
[[https://colab.research.google.com/drive/1XMSfX646Gj-83531zYt1wXZ8RA0N1uOP?usp=sharing|Click here for the report sample]]

===== - Introduction and theory =====
{{https://upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Forces_acting_on_a_simple_pendulum.svg/807px-Forces_acting_on_a_simple_pendulum.svg.png?nolink&500}}

For a simple pendulum

$$ T = 2\pi \sqrt{\frac{L}{g}} \Rightarrow g = 4\pi^2 \frac{L}{T^2}. $$

For a compound pendulum

$$ T = 2\pi \sqrt{\frac{\frac{K^2}{l}+l}{g}} $$

and, hence, a compound pendulum is equivalent to a simple pendulum if

$$ L = \frac{K^2}{l} + l \Rightarrow l^2 - lL + K^2 = 0 $$

which is a quadratic equation with two solutions $l_1$ and $l_2$ where $l_1+l_2=L$ and $l_1l_2=K^2$.

You have to find gravitational acceleration $g$ and radius of gyration $K=\sqrt{l_1l_2}$.

===== - Method and data =====

==== - Data table ====
^ Hole no. ^ Distance [cm] ^ Trial ^ Time for 10 oscillations [s] ^
| 1 | 10 | 1 |  |
|:::|:::| 2 |  |
| 2 | 20 | 1 |  |
|:::|:::| 2 |  |
| 3 | 30 | 1 |  |
|:::|:::| 2 |  |
| 4 | 40 | 1 |  |
|:::|:::| 2 |  |
| 6 | 60 | 1 |  |
|:::|:::| 2 |  |
| 7 | 70 | 1 |  |
|:::|:::| 2 |  |
| 8 | 80 | 1 |  |
|:::|:::| 2 |  |
| 9 | 90 | 1 |  |
|:::|:::| 2 |  |

===== - Graphical analysis =====

===== - Calculating g =====
Find $R$ from here: https://rechneronline.de/earth-radius

$$ g = \frac{GM}{R^2} $$

===== - Discussion and conclusion =====
Answer the following questions in Discussion.
  - Why the angle of deflection of the pendulum should not be large?
  - Why are the periods at 10, 30, 70 and 90 cm similar?
  - Why do you get two symmetric curves after plotting $T$ as a function of $l$.
  - If your were periods at 10, 30, 70 and 90 cm were not as similar as expected, discuss why this happened?