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    Could someone please point me in the direction of some material that can explain a method of calculating the equation of any line. I'm doing some coursework on simple harmonic motion and pendulums for physics in which I am simply changing the length of the pendulum and recording the time period. I have now graphed my results and the line should have the equation y=2pi*root(x/9.8) but I need to be able to prove this.
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    Rather than finding an equation for the line you plotted,you can plot the equation you're supposed to get, and compare with your results
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    The whole point of this experiment though is to show how you could arrive at that equation without knowing it beforehand
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    Unless you are using some knowledge about the physics of the experiment and have an expected relation, it won't be possible to look at a curve and come up with an equation from scratch. You won't know whether it's square root graph, parabola, cubic or anything else.
    However, if you have straight line you know that the form has to be y=mx+c, and you can find the gradient and y-intercept (if there's one) from your plot. In order to get a straight line, you should plot y against some power of x. You would normally know what power, looking at the expected equation, but if you don't want to use that at all, you can try x^0.5, x^2, etc and see which ones gives a straight line.

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    (Original post by Jordan97)
    Could someone please point me in the direction of some material that can explain a method of calculating the equation of any line. I'm doing some coursework on simple harmonic motion and pendulums for physics in which I am simply changing the length of the pendulum and recording the time period. I have now graphed my results and the line should have the equation y=2pi*root(x/9.8) but I need to be able to prove this.
    Note that this result is valid only if the pendulum is displaced by small angles, where 9.8ms^-2 is the acceleration due to gravity(g). When the pendulum is displaced by a small angle, say #, then, you have mgcos# working in a the same line as the string of the pendulum, and mgsin# perpendicular to it (m is the mass of the bob). Thus, the acceleration towards the centre of motion of the bob is -mgsin#/m, or, -gsin# (Don't forget the - sign). Now, lets say that the horizontal displacement of the bob is h (which is approximately equal to the real displacement of the bob, as the angle # is very small). Thus, sin# is approximately h/x (x is the length of the string). Thus we have the acceleration as -gh/x. I hope you know how to obtain the time period if the acceleration is given.
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    (Original post by PhysicsMathsTut)
    Unless you are using some knowledge about the physics of the experiment and have an expected relation, it won't be possible to look at a curve and come up with an equation from scratch. You won't know whether it's square root graph, parabola, cubic or anything else.
    However, if you have straight line you know that the form has to be y=mx+c, and you can find the gradient and y-intercept (if there's one) from your plot. In order to get a straight line, you should plot y against some power of x. You would normally know what power, looking at the expected equation, but if you don't want to use that at all, you can try x^0.5, x^2, etc and see which ones gives a straight line.

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    Ahhhh that's an idea! I think I will try that, thank you


    (Original post by Spandy)
    Note that this result is valid only if the pendulum is displaced by small angles, where 9.8ms^-2 is the acceleration due to gravity(g). When the pendulum is displaced by a small angle, say #, then, you have mgcos# working in a the same line as the string of the pendulum, and mgsin# perpendicular to it (m is the mass of the bob). Thus, the acceleration towards the centre of motion of the bob is -mgsin#/m, or, -gsin# (Don't forget the - sign). Now, lets say that the horizontal displacement of the bob is h (which is approximately equal to the real displacement of the bob, as the angle # is very small). Thus, sin# is approximately h/x (x is the length of the string). Thus we have the acceleration as -gh/x. I hope you know how to obtain the time period if the acceleration is given.
    I'm assuming it would just be a simple SUVAT equation now but why is the acceleration negative?

    Also, how small angles are we talking about? If this is only true for very small angles then it's going to become near impossible for me to measure them
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    (Original post by Jordan97)
    Ahhhh that's an idea! I think I will try that, thank you




    I'm assuming it would just be a simple SUVAT equation now but why is the acceleration negative?

    Also, how small angles are we talking about? If this is only true for very small angles then it's going to become near impossible for me to measure them
    Acceleration is negative, well umm, consider than, you are displacing the bob from the equilibrium position, then the velocity of the bob steadily goes on decreasing. As for small angles, ideally, the angle should be infinitesimally small, but I guess that you should be able to be obtain a fairly close result for angles upto 5 degrees.
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    y-y1 = m(x-x1)

    y1 is the y coordinate of a point on the line and x1 is the x coordinate of the same point. m is the gradient
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    \begin{aligned} y &= ax^n \\ \log (y) &= \log (ax^n) \\   \underbrace{\log(y)}_{Y} &= \underbrace{n}_{m} \underbrace{\log (x)}_{X} + \underbrace{\log (a)}_{c} \end{aligned}

    if you suspect a square root relationship, you can plot log(y) and log(x) against each other and you should get a line with 0.5 as the gradient
 
 
 
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