Determining the acceleration due to gravity using a freefall method

I'm going to do an experiment to measure the acceleration due to gravity in freefall this is required practical for most examples in year 12 I've been given permission by dr. John none of the National Physical Laboratory to use his very fine virtual physics laboratory simulation package to to show you this experiment I'm very grateful to him for that you can find details about that on the website and if you look on the video just over there you can see the website there so in this experiment we're going to drop ball between two light gates and as it passes through the first light gate it's going to start timer and as it passes through the second light gate it's going to stop the timer to make sure we get nice clean release of the ball we're going to use an electromagnet up here where when switch this switch it temporarily stops the electromagnets and so the steel ball will be released the equation we're going to use here is one of these so-called suvat equations which is that the displacement of body if it's experiencing uniform acceleration is going to be equal to the initial velocity of the body multiplied by the time that is been moving for plus half of the acceleration of the body multiplied by the square of the time it's been moving for and remember the acceleration must be uniform for that equation to work now I'm going to be measuring two quantities I'm going to be measuring the distance that the ball is falling which is displacement should say downwards displacement of the ball and I'm going to be measuring the time it takes to fall between those two light gates and that's the here so these are the quantities I'm I'm going to be measuring the displacement of the ball is the independent variable that's the one I'm going to choose to change so which is an in this equation so I'm going to change that variable by varying the distance between the two light gates and you can see here that I'm able to do that by moving either of these light gates up and down now is the initial my apologies you is the initial velocity of the ball as it passes the first light gate and that's because is time measured between the two light gates and so because want to keep you as constant then need to make sure this top light gate doesn't move so as vary the distance between the light gates will only do that by moving the bottom light gate let's first of all get our measurements and then we'll look at what we can do with them so you can see I've got table I've drawn just down there where I'm going to fill in my results and let us start with small distance let us start with 10 centimeter displacement of the ball you can see the resolution of this ruler is 1 centimeter and so can't write down number with any more precision than then have on that table so let us drop the ball and see what happens light gates are powered on release the ball and we have time of 0.06 seconds now we should take repeats but that this is simulation the likelihood is that it will always give us the same number let's just try it again and see what happens 0.06 seconds 0.06 seconds ok so there really is no pointers with the simulation taking repeats but in reality we would take repeats so let us change the displacement now and we're going to change the displacement to 20 centimeters and then we're going to release the ball so release the ball and we get point 1 0 seconds and notice that we've recorded that as 0.1 0 we're not being lazy of 19.1 because that way would be shaving off one of the significant figures and we do need that the number of decimal places in our table indicates the precision of the measurement and the precision of the measurement depends on the resolution of the instrument now as the resolution of the instrument here is given two point zero one seconds for the light gate for the timer and to the nearest centimeter for the ruler then need to make sure the precision of the number write down in the table is matching that resolution okay let's change that distance throw that displacement again let's now make it 30 centimeters between this notch and this notch and we'll release the ball and we get 0.14 seconds and let's do this again so reset put the ball back up there let's increase this to 40 centimeters release the ball and we get point one seven seconds and you'll see this practical really doesn't take very long to do let's go for another 10 centimeters so that's it this as close as like close as can and so this is 50 centimeters release the ball point to zero seconds and we've reset it and we move to 70 centimeters we'll make this our last measurement and we release the ball and we get point two six seconds now why do we take seven measurements not because seven is particularly auspicious number but it's because if you take averages of repeat readings of the same measurement it turns out statistically that if you take seven readings the way of estimating the error in that list of numbers as half of the range gives you as close as possible to the more statistically rigorous way of finding the error which is to do the standard deviation divided by the number of measurements minus one think that's correct so there's very statistically rigorous way of finding an error and there's the level very quick method of let's just take half of the range and obviously hope you can appreciate that if you take lots of measurements the standard deviation and standard error from the standard deviation is going to decrease but with the very quick level way of estimating errors as half of the range more measurements increases the chance of that range actually increases so whilst you have to know that taking more measurements reduces the error practically when you make quick estimate of an error in level physics taking repeats actually increases the error and it turns out that if you do the experiment yourself and you take lots of measurements and you do it for all of the 12 required practicals at level which I've done I've done this quick test there in my experience from my results 11 out of the 12 practicals the most similar errors by the approximation and by the more statistically rigorous method was when you took seven results if you took an eighth result then the error from the statistical method dropped bit too low and if it took too few results okay it was too high so and that's why we pick seven results when you go on to do your degree in physics that's half of the range estimate of the error goes out of the window and I'm afraid you'll have to work out errors properly but as there's far too much maths to include in your level physics as it is we can't even do calculus vectors aren't done very well in a-level physics that are better in little maths and so because of that I'm afraid errors and statistics that goes into errors that had to take backseat now let's look at that equation again we've got our results but let's look at our equation again we had displacement is equal to initial velocity multiplied by time plus half of the acceleration which is constant multiplied by the square of the time and we want to find what is so we need to get this equation in the form equals MX plus so we can extract and so what we've got here is variable and we've got variable and we've got variable and I'm going to color my variables with this orange highlighter and we've got constant and we've got constant and I'm going to color my constants with the blue highlighter and we need to get this equation the form equals MX plus so that's going to be variable equals let's just change my pen to black one equals constant times variable so it's constant times variable plus constant and that's the form equals MX plus now in other parts of the world they do use different form of this in the United States of America for example believe they use equals if can get this right plus where is the gradient and is the y-intercept here in the UK we use equals MX plus where is the gradient and is the y-intercept but either way around the equations in the same form it is variable equals constant times variable plus constant and so this equation is not yet in this form so to get it in this form what we're going to do is divide through by if divide through by get the displacement divided by the time equals the initial velocity plus half of the acceleration multiplied by the time and let's just color in our variables so all of this is variable because was variable and was variable over is variable and this over here is our variable which is and our constants are our color in blue was our initial velocity and our 1/2 of the acceleration and really hope you can see that that is in the same form as the as these spaces but beneath them all I've got to do is move some things around bit and I've got exactly the same equation and I'll do that now so if plot on my y-axis over and on my x-axis then my gradient will be half and my intercept will be and so just going from one equation dividing through by t2 force it into the form equals MX plus or if you're different part of the world like was plus BX and you can you can now see what you have to plot and what we're going to plot is graph where on the axis we will have over and on the axis we will just have now our displacement is going to be downwards our acceleration is down our initial velocity when we passed that first light gate is downwards so let's just pull downwards the positive direction and so is going to be positive so we would expect line straight line of acceleration is uniform and that's what we've said we're hoping is the case so we're expecting straight line which passes through the y-axis at an initial velocity and has gradient when we calculate it gradient which will be equal to half well we want to find what is because is the acceleration due to gravity so what we're going to do is we are going to do twice the gradient once found what the gradient is shall double it and that will be my acceleration due to gravity so I've taken those results already and I've put them into spreadsheet and anyone that's taught by me will know that do enjoy spreadsheets now your lots probably more than should and so we wanted to get this into the form over so need to divide by divided by just put the equation down there so you can see it and you can keep up with what we're doing so I've got my distance I've got my time in my spreadsheet little thing about spreadsheets like Excel other spreadsheets are available it does some funny rounding things here but you're going you can go into here and you can you can change the precision of that number you'll be click on home and in Microsoft Excel then you click on number and then you can you can just easily change the precision one should be able to easily change the precision that way let's see this is going to play ball yes there we go so we want that precision we've got distance in centimeters why not change it into meters already because there really is no advantage to keeping it in meters in centimeters so let's just do our formula and hope that text is appearing okay so equals that divided by hundred that's our distance in meters of course the precision of that numbers ROM so need to go into here select the numbers and increase the precision of that number again oops wrong way and but there we go there's there's my distance now what need is distance over time shouldn't pull this distance should should call this displacement very clumsy of me let me go ahead and change these to displacement displacement okay so now what are these do is get my displacement divided by my time or my over and that's going to be equal to this cell divided by this cell and there we go there's my values for my displacement divided by my time now how many significant figures should you write this number two this is very controversial point because the numbers I've used in my calculation are two significant figures it sounds like should match the number of significant figures here but of course if did that then I'm not being consistent with decimal places and suppose does it matter and the good news is that AQA in particular this year decided that or not this year so with this new a-level decided that they're going to be less fussy about that so if you want to keep an extra significant figure so that you're consistent with decimal places that's fine and would always encourage you to go up in terms of precision rather than down for calculated number because you can always round your final answer anyway so now what need to do is plot graph of over versus and if select my cells like this can slide my cells like this can go into insert can go to charts and want to choose scatter well don't want to choose line even though you think you want line graph I'm gonna choose scatter and the reason for that is we choose line it has category axis now quick check to make sure I've got my axes the right way around over 2 and the axis that is correct and on the axis that is correct so there's my graph now want to can mess around with the formatting and make this graph of it prettier maybe I'm going too make this on page all of its own so move chart new sheets and give it sensible name and all the rest of it but I'll do some quick formatting now okay so did little bit of formatting I'm happy with those data points now can add my line of best fit which they call trendline want straight line and if want can extrapolate that back by going down to here it says backwards and if go backwards that's little bit too far but can always just change my axis so that the minimum is zero and the axis on the so that the minimum is zero and then don't have to worry about what the period means in the in the casting the trend line back because he is strange dark heart now if you look at this it doesn't necessarily look like our trendline is actually straight line and it should be but it doesn't look like it is so what does this tell us this probably tells us that our assumption the acceleration was uniform is not right it's almost straight line we've got some points above and below it's almost straight line so think it's good enough so how do we find the gradient well there's rather crude ways of doing it for example we can format trendline and we can ask to show an equation and then we get the gradient appearing here and let's just make the fonts so it's readable we get the equation of the straight line here and hopefully you can see that's in the form equals MX plus and if you remember we were interested in the gradient and double the gradient was our acceleration due to gravity this is coming in at about ten point one sorry ten point two ish ten point two meters per square second which is pretty good but there's more elegant way of doing it which is to come into here and type equals slope open brackets and then select all of your values which is these and all of your values which is these and this gives you your value of the gradient and we know that twice the gradient is our acceleration due to gravity and that is the experiment for finding the acceleration due to gravity from from freefall please do like and subscribe 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