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--- courses:phy101l:4 [2023/05/27 03:53] – created asad
+++ courses:phy101l:4 [2023/11/05 02:37] (current) – [1. Introduction] asad
@@ Line 1: / Line 1: @@
 ====== 4. Spring constant from extension and period ======
+[[https://colab.research.google.com/drive/1tD3YzjG3QRVFyrV-L1ccF84yx7QW4eYf?usp=sharing|Google Colab example]]
+===== - Introduction =====
+Spring constant is a property of a spring; its value $k$ should be a constant. You will calculate $k$ using two different methods: first, using the extension $l$ caused by a hanging mass $m$ and second, using the period $T$ for a given hanging mass $m$.
+{{:courses:phy101l:spring.png?nolink|}}
+When a mass $m$ is hung from an unstretched spring, it is extended by a length $x=l$ because of the gravitational pull of the earth on the mass. The spring exerts a restoring force $F$ on the mass opposite to its gravitational force $mg$. According to Hooke's law
+$$ F \propto -l \Rightarrow F = -kl $$
+where $k$ is the spring constant. Replacing $F=-mg$ we get $-mg = -kl $ and
+$$ k = g\frac{m}{l}. $$
+$$ l = \frac{g}{k}m + 0 $$
+For the second method, you will use the relation between period and mass
+$$ T = 2\pi \sqrt{\frac{m}{k'}} $$
+which leads to
+$$ k' = 4\pi^2 \frac{m}{T^2}. $$
+$$ T^2 = \frac{4\pi^2}{k'} m + 0 $$
+The values $k$ and $k'$ should be very similar because they are both the spring constant of the same spring.
+===== - Method and data =====
+^ Mass $m$ [g] ^ Extension $l$ [cm] ^ Time for 10 oscillations $t$ [s] ^
+| 100 |  |  |
+| 150 |  |  |
+| 200 |  |  |
+| 250 |  |  |
+| 300 |  |  |
+===== - Spring constant from extension =====
+===== - Spring constant from period =====
+===== - Discussion and conclusion =====
+  - Why are $k$ and $k'$ different?
+  - Which one is greater, $\delta k$ or $\delta k'$? Why?
+  - In which method we have higher fitting error?