25 February 2015. Determining Gravity from a Free Fall Experiment (using excel)

Purpose
to examine the statement that states: "In the absence of all other external forces except gravity, a falling body will accelerate at 9.8 m/s²", by analyzing the motion of free-fall body.

Apparatus:

1. A set of a heavy tripod base with leveling screws, a fee-fall body, a weighted clip to anchor the spark paper, and an electromagnet with power supply as shown in the picture. This device is purposely used to observe the motion of a free-fall body.
   
   
   
   
2. A spark generator: a generator of electric oscillations that utilizes the discharge of a condenser through a spark gap as the source of its alternating-current power. In this experiment, the frequency of the spark is 60 Hz. That means it will produces a dot every 1/60 s.
3. A tape and a ruler (metric units).

Procedure:

The Professor will do the step 1-7.

1. Turn the dial hooked up to the electromagnet up a bit.
2. Hang the wooden cylinder with the metal ring around it on the electromagnet.
3. Turn on the power on of the spark generator.
4. Hold down the spark button which will generate 60 Hz frequency and leave dots on the paper.
5. Turn the electromagnet off, so that the spark generator fall.
6. Turn off the power of the spark.
7. Tear off the paper strip to examine the series of dots found.

Data Analysis:

• Notice that we have series of dots on the paper corresponding to the position of the falling mass every 1/60 s.

• Then we measure the length of each dots using the metric ruler.
• In Excel, we compute the time, which is added by 1/60 s every time we counter the next dot, in comparison with the length from one dot to another. (Shown in the picture below)

• Next, we compute difference in distance (∆x), which is X-two - X-one; the mid-interval time which is t-two - t-one divided by 2; and the mid interval speed (V) which is the ∆x/(1/60).

• Then, we graph "Distance versus Time" and "Velocity versus Mid-Interval Time".

• Now, from logical thinking, we can see that from the "Velocity vs mid-interval time" that they have a linear relationship, which means that the slope (the acceleration, since accelerations is the derivative of velocity) is constant. Therefore, the instantaneous velocity in the middle of any time interval is the same as the average velocity because the average value of a straight line is always at its middle.
• We can also prove this by calculation as shown below.

• Next, from the graph "Distance vs Time" we can find our value of acceleration by differentiating twice.

• From the graph "Velocity vs mid-interval time" we can find our value of acceleration by differentiating once.

"Errors" in Data Analysis:

• As we can see from the calculation above, we can see some "errors" in comparison with the "true" value of gravitation (as shown below).

• Now, we are going to analyze the class data for the value "g", using excel to calculate the standard deviation of the mean.

• The standard deviation of mean that we got is 0.201. Now, if our data are properly distributed according to the graph below, we can conclude that:
• If we say g= 9.56 +/- 0.201, we can state that we are 68.26% sure that it is true,
• If we say g= 9.56 +/- 0.402, we can state that we are 95.44% sure that it is true, and
• If we say g= 9.56 +/- 0.603, we can state that we are 99.74% sure that it is true.

Conclusions:

• Our pattern of the values of the calculated g are basically randomly distributed around the average value of g.
• Our average value of g in comparison with the true value of g as had been calculated above is:
• 0.1879 in average absolute differences, and
• -1.914% in average relative differences (also called percent error).
• Random error by definition is caused by any factors that randomly affect measurement of the variable across the sample. It will not affect the average (since the amount of positive errors and negative errors are the same, they will cancel each other out), it just adds variability. In this experiment, the random errors could be the "guessing" part of the last decimal places of the length between dots is not really accurate (it is different between each students). 
• While systematic error is caused by any factors that systematically affect measurement of the variable across the sample. Since all equipment  used in this experiment are adequately precise and accurate, there is likely no systematic error that occurred.
• While we are trying as hard as possible to gain a perfect result with no errors, in reality there will always be errors caused by humans directly or indirectly. Nevertheless, with good apparatus, we managed to gain a fairly small deviation of 0.201 which is quite satisfying and enough. Just imagine if the sparker is less precise, with 0.1 s delay every time it sparks a dot. It would really affect the whole calculation and we will be way off from the true value of gravitation at the end. Therefore, good precise apparatus, which is also expensive, really help us in avoiding unwanted errors.