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I want to determine the acceleration due to gravity at a place using a helical spring.
For this, I've first calculated the extension in length of the spring (##x##) for a particular load (##L##) on the pan. Then I've plotted a graph for ##L## vs ##x## for different values of L and corresponding values of ##x##, and calculated it's slope.
Next, I've calculated the time period for 20 oscillations of the spring for each of the weights I've used before. I've calculated the time period for one oscillation, and taken the average of 3 readings (##T##). I've plotted a graph for ##L## vs (##T ^2 ##), and calculated it's slope.
I've used this formula for the calculation of acceleration due to gravity: $$g= 4 \pi ^2 \frac {Slope \; of \; L \; vs \; T^2 \; graph}{Slope \; of \; L \; vs \; x \; graph} $$ which I derived from the formula $$ T = 2\pi \sqrt {\frac {x}{g} } $$ from where we can get $$\begin{align} g = 4 \pi ^2 \frac {x}{T ^2} \nonumber \\ &= 4 \pi ^2 \dfrac {\frac {L}{T ^2}}{\frac {L}{x}} \nonumber \end {align}$$.
Are these formule correct? I've got the value of ##g=10 m/ s^2##, which seems to be approximately correct, but I wanted to know whether the process I used is correct.
For this, I've first calculated the extension in length of the spring (##x##) for a particular load (##L##) on the pan. Then I've plotted a graph for ##L## vs ##x## for different values of L and corresponding values of ##x##, and calculated it's slope.
Next, I've calculated the time period for 20 oscillations of the spring for each of the weights I've used before. I've calculated the time period for one oscillation, and taken the average of 3 readings (##T##). I've plotted a graph for ##L## vs (##T ^2 ##), and calculated it's slope.
I've used this formula for the calculation of acceleration due to gravity: $$g= 4 \pi ^2 \frac {Slope \; of \; L \; vs \; T^2 \; graph}{Slope \; of \; L \; vs \; x \; graph} $$ which I derived from the formula $$ T = 2\pi \sqrt {\frac {x}{g} } $$ from where we can get $$\begin{align} g = 4 \pi ^2 \frac {x}{T ^2} \nonumber \\ &= 4 \pi ^2 \dfrac {\frac {L}{T ^2}}{\frac {L}{x}} \nonumber \end {align}$$.
Are these formule correct? I've got the value of ##g=10 m/ s^2##, which seems to be approximately correct, but I wanted to know whether the process I used is correct.
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