# Practical: Measuring g (falling ball)

### Introduction

In this practical the acceleration due to gravity will be calculated by using an electronic timer to measure the time taken for a small steel ball to fall a known distance.

### Method

The apparatus is set up as in the diagram. Find out how the release mechanism works and make a couple of trial runs to see if it works properly. Measure the time for the same height and see if you get the same reading.

Since the acceleration of the ball is constant we can use the formula s=ut +½at^{2} to represent the motion. The initial velocity is zero so this simplifies to s=½at^{2} where s is the distance fallen, a is the acceleration due to gravity and t the time taken. By taking appropriate measurements use a graphical method to show that s=½at^{2} and determine g.

### Collecting raw data

Prepare a table in excel, like the one below, ready to receive your results. You will need to estimate the uncertainties in your measurement of height and time, these should be entered into the headers of the table.

- The uncertainty in height depends on how well you think you can measure the position of the ball with the ruler, if you think of any neat way to make it better then do it and make a note of what you did.
- The uncertainty in the timer can be taken to be the last digit of the reading (e.g. if the reading is 0.236s then the uncertainty is 0.001s). Enter this value in the header of each time measurement. However the uncertainty in your method is probably greater since it is impossible to do everything exactly the same many times. To find this you will repeat the measurements several times and use the spread of data to determine the uncertainty.

Fix the height and release the ball, enter the values into the table then repeat the process 4 more times. Change the height and do it all again until you have measurements for about 8 different heights.

### Processing the Data

The first stage in processing is to calculate the average value of t and the uncertainty in each run.

- Add two columns to you excel table as shown below.
- Add equations to these columns to calculate the average value of the time =AVERAGE(B2:F2) and the the uncertainty in the values =(MAX(B2:F2)-MIN(B2:F2))/2 . The formulas can be copied into all the cells by dragging the bottom right hand corner of the cell downwards.

As mentioned earlier the equation for the motion of the ball is s=1/2at^{2} , this means that s is proportional to t^{2} so a graph of s vs t^{2} will be a straight line with gradient 1/2a. However in this experiment you have been changing the height (s) and measuring time (t) so height is the independent variable and time the dependent. In this case you should plot t^{2} on the y axis and s on the x axis which will give a graph with gradient 2/a (t^{2}=2s/a). Before this graph can be plotted the data needs to be processed further.

- Add four more columns to the spreadsheet as shown below.
- In the "average time
^{2}" column add the formula =G2*G2 and copy it down by dragging the bottom right hand corner. - Calculate the max and min time
^{2}by inserting the formulae =(MAX(B2:F2))*(MAX(B2:F2) ) and =MIN(B2:F2)*MIN(B2:F2) and copying down. - The uncertainty in time
^{2}is found from the 1/2 the difference between the max and min, in this case that would be =(J2-K2)/2

### Presenting Processed Data

You are now ready to draw a graph of t^{2} against s.

- Copy the t
^{2}and s columns into the graphing programme Loggerpro. - Arrange the graph so that t
^{2}is on the y axis and s on the x axis. - Add an extra column to the table and enter the uncertainty in t
^{2}data, use this column to plot the error bars for t^{2}as you have don before. - Plot the best fit straight line and manually place the steepest and least steep lines.
- Calculate the acceleration of the ball from the gradient of the best fit line.
- Calculate the uncertainty in your final result from by using the steepest and least steep line (the uncertainty is given by 1/2(max value-min value)).

### Conclusion and Evaluation

Compare your result with the accepted value.

- Within the level of uncertainty quoted is your value acceptably close to the accepted?
- If your answer is further away from the accepted value than the uncertainty can you think of any reasons why?

Look at your graph and consider the following questions:

- Was the acceleration of the ball uniform?
- Did the error bars truly reflect the spread of data?
- Is the intercept on the y axis 0? If not what might have caused it not to be?
- According to your results what was the weak point in the experiment?
- What could be done to address the weaknesses mentioned?