Freefall lab results (2017)

Data

The average heavy object was about 26 times the weight of the average light object.

Here's (most of) your raw data

Keep all the data?

  • The horizontal axis is just the number of the measurement.
  • The vertical axis is the measured accelerations, in $m/s^2$.

Draw and label a horizontal line at the class average acceleration for the light objects. Do the same for the heavy object average.

Do those look like they are "in the middle" of the light/heavy object data?

Based no the averages, do the heavy or light objects look like they have the greater acceleration?

Averages

I'll show some diagrams from a previous class. You will do some things along the same lines with your class data. The average 2015 accelerations were light objects: -8.46 $m/s^2$; heavy objects -8.97 $m/s^2$.

Which kind of object appears to fall with a greater acceleration based on the averages?

Uncertainty

Repeated measurement of the same thing does not always give the same result.

There is a standard way to quantify the uncertainty in a set of repeated measurements. Here goes...

The brown bars are the "deviations" of each data point from the average.

The overall uncertainty seems like it should be related to some sort of "average deviation". But if you literally take the average of the (positive or negative) deviations, you will get 0, so that's no good.

The most common way to quantify the uncertainty is to calculate something called the standard deviation, "$\sigma$":

  • Find the deviation for each data point,
  • Square it.
  • Find the average of those squared deviations.
  • Take the square root of that average.

This quantity is call the standard deviation, which is written as $\sigma$ ('sigma').

In a statistics course you would find that for typical experiments ("Gaussian distribution of errors"), if you measure a quantity $x$ many, many times, 68% of the measurements will be between $x_{\text ave}-\sigma$ and $x_{\text ave}+\sigma$.

I'll round this and call it the "70% rule".

Another way to put this: 70% of the time, the "real value" of some physical quantity will be found between $x-\sigma$ and $x+\sigma$.

For the 2015 light object data, $\sigma=0.41 m/s^2$.

The length +$\sigma$-(-$\sigma$)=2$\sigma$ is usually the size of the "error bars" on a graph.

Your data - Mark horizontal lines at $\sigma$ units above and below your average for the light data. That is, at -9.03$\pm$1.09.

70% of your data should lie within that range. Let's see... 70% of 10 datapoints is 7 points. Is that about right for our class data?

Finding the standard deviation *graphically*

In this course you will *never* calculate standard deviations by calculating the sum of the squared deviations, but instead...

You can estimate $\sigma$ visually from your data, using the "70% rule".

Do this with your heavy object accelerations:

  1. Count the number of measurements you have, and figure out 70% of the number of measurements you have made. (And also what is 30% of the number of measurements.)
  2. Draw lines around your measured values such that (a) about 30% of the points fall outside of the lines, and (b) there are just as many points above both lines as below both lines. (Now your lines should enclose about 70% of your data).
  3. The distance from your lower boundary to your upper boundary is $2\sigma$. So divide the distance by 2 to get $\sigma$.
  4. To get the average, by eye pick out a position which would be half-way between the two lines. This is your average value.

Compare your visual estimates of $\sigma$ and average value with what the spreadsheet calculated.

Are the measured accelerations *that* different??

Time to look at the data again comparing heavy and light objects (together with variation). For the heavy objects, $\sigma=0.69$.

The criterium for measurements of two different quantities to be considered to be considered statistically "different" is that the error bar ranges of each quantity should not overlap.

Do your heavy and light objects have "different" accelerations by this criterium?

What would you expect?

After many, many experiments, of this sort, what do professional scientists think?...

Falling objects and GravE

Talking about lifting books onto shelves...

  • To lift a book you need to "do work" on it, equal to $w=F_g /Delta h$.
  • We asked "where does the energy go"?:
  • Answer: $$\Delta \text{GravE}=mg\Delta h.$$

Now, what if we drop the book off the shelf.

Every second that an object is falling it is losing GravE.

Where does that energy go???

Hint: what is changing while the object is falling?

Image credits

The very honest man