Introduction

In this practical the acceleration of a ball rolling down a slope will be determined by measuring the time it takes to roll different distances.

 

Method

The metal rulers we have in the lab have a convenient groove in them so can be used as a track for a rolling ball.

Make the ruler into a slope by lifting up one end about 5cm. You are going to measure the time for a ball to roll down the slope using a stop watch so it helps if you place something metallic at the end of the slope, you will then be able to hear when the ball reaches the end.

If we assume that the acceleration of the ball is constant then the displacement s is related to the time, t by the equation

s= ut +½at²

The initial velocity = 0ms^-1 so this simplifies to s=½at²
Use a stop watch to measure the time for the ball to roll different distances then use a graphical method to determine the acceleration, a of the ball

If this practical was going to be assessed the following information would not be given however with this information the practical would be a good introduction to processing of data as well as what is required when writing a conclusion.

 

Collecting Data

To keep the data organised it will be collected in a table. This can be done directly into excel or you can do a rough one on paper then transfer it to excel. The table should have headings as shown below.

 

Independent variable

In this practical the distance travelled down the slope is the independent variable; this means it is the one that you change. You need to have a wide range of different values which will allow you to plot a meaningful graph. If the distance travelled is too short then you won’t be able to measure the time so start at 1m and end at 20cm taking measurements every 10cm.

 

Dependent variable

Time is the dependent variable because it is dependent on the distance travelled. You are going to repeat each measurement 5 times to give some idea of the random variation of the measurements.

 

Uncertainties

It is not possible to measure the distance or the time exactly; we say that all measurements have uncertainties which are quoted as ±.  

• Determine the uncertainty in distance by deciding how well you think you can measure the distance travelled by the ball. Does the shape of the ball affect this?
• It is difficult to determine how well you can use the stopwatch (that’s why you are going to repeat the measurement) but you can give the uncertainty a rough value by quoting the smallest digit. So if your stopwatch measures to 0.01s then the uncertainty is ±0.01s.
• Add uncertainties into the table and write an explanation of why you decided on these values.

 

Taking measurements

You are now ready to take measurements. Observe carefully as you do this and make notes of anything that might be relevant when you come to evaluate the results. If you get a value that is completely different to the others then it can be treated as a mistake and left out.

 

Linearising

The distance and time are related by the equation s = ½at². This means that plotting s against t would give a parabola. Parabolas are difficult to evaluate so it is more convenient to manipulate the data to give a straight line (see Why linearise worksheet).
From the data we can deduce that s is proportional to t² so plotting s against t² will give a straight line with gradient = ½a. This is called linearising and is very common when analysing data from physics experiments.

 

Processing data

So far you have 5 different values of time for each distance. You are going to calculate the average of these values so add a new column to your table.  Excel will calculate the average for you:

• Click the first cell of the new column then type=average (   then highlight the row of times that you want to average. Close the bracket and enter.

The formula can be copied down all the cells by clicking the bottom right corner of the cell and dragging down. You are going to plot a graph of s against t² so need to find t².

• Make a new column and click the first cell.
• Add the formula =G2^2 where G2 is the av. time that you just calculated. You don’t need to type G2 you can simply click the cell and G2 will appear in the formula bar.
• Copy the formula down as before.

 

Processing uncertainties

The uncertainty in t² is not the same as the uncertainty in t. To find out what it is you halve the difference between the maximum value squared and the minimum value squared.

• Add another column with the heading unc. t²/ s²
• Add the formula = =(MAX(B3:F3)*MAX(B3:F3)-MIN(B3:F3)*MIN(B3:F3))*0.5 in the first cell of the column.
• Copy the formula down as before.

• The uncertainty is just an approximation so can be quoted to 1 significant figure. This is done by highlighting the cells then right clicking and choosing > format cells > number > decimal places 1.
• It doesn’t make sense to quote t² as 6.1504 if the uncertainty is ±0.2. Make the data consistent by setting the number of decimal places in t² to the same as the uncertainty (e.g. 6.2) set the number of decimal places by formatting the cells as before.

 

Presenting data (in a graph)

• The graph can be drawn in excel but the program “LoggerPro” has some neat features that make it better so this is what you are going to use.
• Open the program LoggerPro
• Copy and paste the columns “distance” and “av.t²“ from excel into the blank table in LoggerPro.
• To add headers in double click the table header and fill out the form. Do this for all 3 columns.
• The graph is drawn automatically but it might be too small to see. Zoom in on the graph by clicking the auto zoom button.

• To plot the best fit line click the best linear fit button, the gradient and intercept will be displayed.
• To move around the graph axis place the cursor near the axis until you get a wiggly arrow, click and drag to see what happens. There are three different wiggly arrows on each axis, find out what they do.
• You can change the axis by clicking on the labels and choosing the quantity you want.

 

Error bars (s)

To add error bars to the distance values double click the distance header and choose the Options tab. The uncertainty in s is the same for each value so tick “error bar calculations” and “Error constant”. Enter the uncertainty in the box.

 

Error bars (t²)

The uncertainties in t² are in the 3^rd column. Double click the t² header but this time choose “Use column” and select the unc. t² column.

Now you have drawn the error bars check that the best fit line passes through all of them. If it doesn’t then you can’t confirm that s is proportional to t² so you might as well stop here. If this condition is satisfied then you can go on to find a value for a.

 

Finding a

The whole point of drawing the graph was to find the acceleration of the ball. The equation of the line is s=½at² so the gradient is ½a. Use the graph to find a value for a.

 

Uncertainty in a

To quote the uncertainty in our final result we can draw the steepest and least steep lines that touch all of the error bars.  You can place lines by hand by clicking the “curve fit” button. This opens the pop up window below.

• Select “Linear” from the list of equations on the left.
• Click “try fit”
• Select manual at the top right hand corner.
• This will be the best fit line, to make the steepest line use the + and – buttons to the right of the slope and y- intercept to move the line around.
• If the line jumps too much change the increment by clicking the arrow to the right of the + and – buttons.  You can see how this is done on the screencast.
• Move the line so that you are satisfied it is the steepest line through the points. Here it might be simpler to make the line pass through the ends of the first and last error bars, this is acceptable but might not hit all the error bars.
• Click OK then repeat the procedure to get the least steep line.

The uncertainty in the gradient is ± ½(max gradient-min gradient) quote your gradient with its uncertainty.

 

Writing the report

There are 3 parts to the data collection and processing part of the report.

Raw data

This is the table of s and the five runs of t including units and uncertainties plus an explanation of how you came to decide on the uncertainties. The number of decimal places in the data should be the same as the uncertainty.

The uncertainty in distance is ±0.002 due to difficulty of deciding where the edge of the ball was.
The uncertainty in time was ±0.1s as the stopwatch was only accurate to 0.1s

Processed data

This is the table of processed data and uncertainties. It should also include the independent variable. The data should be consistent with the uncertainties and any calculations should be explained.

The average time is the mean of the 5 runs.
The uncertainty in t² is calculated from ½(max time² – min time²)

3.The graph

The graph should be properly labelled (including units), have error bars, a best fit line and steepest and least steep lines. The gradient should quoted including its uncertainty.

From the text box gradient of best fit line = 0.1744ms^-2
Maximum gradient = 0.2144ms^-2
Minimum gradient = 0.1444ms^-2
Uncertainty in gradient = ½(max – min) = 0.035 ms^-2

Gradient = 0.17 ± 0.04 ms^-2
 

 

Conclusion

The aim of this experiment was to find the acceleration of the ball so it should be quoted here with uncertainties.

The gradient = ½a so a = 2x gradient
a = 0.34 ± 0.08 ms^-2

It is not enough just to state this it must be justified, one way of doing this is to compare with a value obtained from a book or in another way. In this case we can say that the acceleration = gsinϑ where sinϑ = 5/100. So a = 0.49 ms^-2 . So the value obtained here was lower than expected even if we take into account the uncertainty the largest value is 0.34 + 0.08 = 0.42ms^-2

Calculate your value for a and compare it with the expected.

Now is the time to comment on the results. The results are presented in the form of a graph so you should comment on that.

How close is the best fit line to the data points?

Do the error bars reflect the actual random error in the points?

Does the line pass through (0,0)?

In the example above the best fit line is close to all but one of the data points. In general the error bars seem a little on the large size as the points are actually quite close to the line (except one). The best fit line doesn’t pass through the origin but if we take the error bars into consideration then it is close enough.

 

Evaluation

This is where you must try to say what it was about the experiment that gave rise to the results achieved.  So if your result was too small as in the example data then you should try to explain what might have caused it to be this way. Depending on how you did the experiment and the nature of your results you will have different things to say. It is important that this stage is completed soon after the experiment so you can remember exactly what you did. For the sample data:

The data shows that there were some fairly big random errors in the time measurement; this could be due to the difficulty in starting and stopping the clock which was done by hand.  It was noticed that the ball did not always run along exactly the same path, sometimes it wobbled significantly. The amount of wobble was related to the way the ball was released and it was found that as the technique was improved the wobble reduced, this can be seen as a smaller spread of t values for shorter lengths. One might expect the uncertainty in the shorter times to be higher but for this reason they are in fact less.  The intercept of the best fit line occurred at -0.031 m, the intercept was supposed to be 0 so this would imply a systematic error in the distance measurement of 3cm. It seems very unlikely that this was the case. This error is more likely to be due to the random error in measurement since the least steep line actually has a positive intercept.

The most likely reason for the difference in acceleration is that the actual acceleration was in fact lower than gsinϑ due to the action of friction and air resistance not taken into account. The fact that the ball rolls is evidence that there is in fact friction between the ball and surface.

 

Improvements

This is where you can suggest ways of improving the experiment. These must address the weaknesses highlighted in the previous section. If the improvements involve changes in the apparatus then they must be explained properly.

The biggest source of error is the measurement of time. This could be improved by using an electronic timer. This would need to be switched on at the top and off at the bottom. To achieve this two light gates could be used. One positioned just in front of the starting position, the other at the end of the track. The time between the ball breaking the first beam to breaking the second would give the time required.  Care would have to be taken to position the light gates accurately so that the middle of the ball always passed through the beam. Positioning of the ball also affects the time as well as the distance, placing an obstruction on the track which was quickly removed would make a cleaner release as well as improving the accuracy of the distance measured.