<![CDATA[
# Arduino Break Beam Gravity
GOAL: Measure acceleration due to Earth's gravity using Arduino microsecond timer circuit
<img width="25%" src="view-20201205_171837.jpg"> <img width="36%" src="viewc-20201205_182222.jpg">
- Drop a ball at L = 10, 20, 30, 15, 25 inches 5x each.
```
O -> ball at L height
|
|
(L-4) inches
|
|
+ -> sensor 1
| -> 4 in
+ -> sensor 2
```
- Measure time in microseconds to fall between sensor 1 and sensor 2
when it breaks the light beam.
- RESULT: my estimate is $g=9.5 m/s^2$. This is within 3% of true
value. Air resistance probably affected my estimate.
- OVERALL RESULT: you can use an Arduino Uno micro-second timer
circuit to accurately estimate acceleration due to gravity.
<video width="36%" controls><source src="breakbeam-measure-20201205_171839.mp4"></video>
================================
# Data
```
The data right of the serial port:
--- Miniterm on /dev/ttyS4 9600,8,N,1 ---
--- Quit: Ctrl+] | Menu: Ctrl+T | Help: Ctrl+T followed by Ctrl+H ---
setup
elapsed_time 54112
elapsed_time 52152
elapsed_time 51488
elapsed_time 51888
elapsed_time 50536
elapsed_time 36364
elapsed_time 35464
elapsed_time 35548
elapsed_time 33984
elapsed_time 36768
elapsed_time 27836
elapsed_time 28036
elapsed_time 27056
elapsed_time 28044
elapsed_time 28972
elapsed_time 39532
elapsed_time 40480
elapsed_time 38668
elapsed_time 41200
elapsed_time 44512
elapsed_time 29268
elapsed_time 30200
elapsed_time 28488
elapsed_time 30252
elapsed_time 29868
```
================================
# What Do the Laws of Physics Say Should Happen?
- $F=ma$ # newton's law
- $a=F/m$
- Force of earth ($E$) gravity at surface on object with mass M. The
acceleration is $F/m = g$ for any mass!
- $F = G m_E m / r_E^2 = g m$
- Knowing the acceleration g we can get the velocity and position by
integrating twice.
- $ v = g t + v_0$
- $x = 1/2 g t^2 + v_0 t + x_0$
- Starting at height L, 0 velocity and triger sensors S apart:
- fall L-S to sensor1 in t1 to v1.
- fall S to sensor2 with initial velocity v1 in time t2
- $(L-S) = 0.5 g t_1^2$ # from x equation
- $t_1 = sqrt(2 (L-S)/g)$ # rearrange: time to get to sensor 1
- $v1 = g t_1$ # velocity at sensor 1 from v equation
- $v1 = g sqrt(2 (L-S)/g)$ # substitute t_1 from a above
- $ v1 = sqrt(2 g (L-S))$ # simplify
----
- $S = 0.5 g t_2^2 + v_1 t_2$ # sensor fall S with initial v1 using x equation
- $0.5 g t_2^2 + v_1 t_2 -S = 0$ # rearrange
- quadratic formula $t = [-b +/- sqrt(b^2 - 4ac)] / 2a$. $a= 0.5 g$. $b= v1$. $c= -S$
- $t_2 = -v_1 +/- sqrt(v_1^2 + 4 \cdot 0.5 g S) / (2 \cdot 0.5 g)$
- $t_2 = -v_1 + sqrt(v_1^2 + 2 g S) / g $ # positive root, time has to be positive
- Substitute v1:
- $t_2 = [-sqrt(2 g (L-S)) + sqrt( (2 g (L-S)) + 2 g S)] / g$
- $t_2 = [-sqrt(2 g (L-S)) + sqrt(2 g L)] / g$
- $t_2 = [sqrt(2 g L) - sqrt(2 g (L-S)) ] / g$
- $t_2 = sqrt(2) [sqrt(L) - sqrt(L-S) ] / sqrt(g)$
- $t_2 = sqrt(2/g) [sqrt(L) - sqrt(L-S) ]$
RESULT: t_2 is the calculated time as measured by sensors S apart
falling from a height of L
================================
# Statistical Data Analysis
Pull the data into R
```
# observed sensor times in microseconds
t = c(54112, 52152, 51488, 51888, 50536, 36364, 35464, 35548, 33984, 36768, 27836, 28036, 27056, 28044, 28972, 39532, 40480, 38668, 41200, 44512, 29268, 30200, 28488, 30252, 29868)/1.0E+06
# the heights the experiment was performed at
l = c(rep(10,5),rep(20,5),rep(30,5),rep(15,5),rep(25,5))
S=4
myx = (sqrt(l)-sqrt(l-S)) # what the linear model inputs. t2 equation
# make a linear estimate of the relationship (least squares: t = A*myx+B)
mylm = lm(t ~ myx)
summary(mylm)
Residuals:
Min 1Q Median 3Q Max
-0.0021731 -0.0007420 -0.0003242 0.0008086 0.0037078
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0001866 0.0012439 0.15 0.882
myx 0.0730061 0.0023870 30.58 <2e-16 ***
# make a linear estimate of the relationship forcing 0 intercept
mylm = lm(t ~ myx+0)
summary(mylm)
Residuals:
Min 1Q Median 3Q Max
-0.0021436 -0.0007521 -0.0003316 0.0008306 0.0037004
Coefficients:
Estimate Std. Error t value Pr(>|t|)
myx 0.0733548 0.0005332 137.6 <2e-16 ***
```
- RESULT: the empirical constant estimate is between 0.0730 and 0.0733:
$t_2 = 0.073*[sqrt(l)-sqrt(l-S)]$
- From model:
- $t_2 = sqrt(2/g) [sqrt(l)-sqrt(l-S)]$
- $sqrt(2/g) = 0.073$
- $2/g = (0.073*0.073)$
- $g= 2/(0.073*0.073)$
- $g= 375.30493526$ inches/s/s
- $g = 9.5 m/s^2$ # 1 m = 39.37 inches
- True: $g = 9.8m/s^2$
RESULT: My estimate of the acceleration due to gravity 9.5 m/s^2 is
within 3% of the true value 9.8 m/s^2. Air resistance?
----
# Plot data
```
# what the model predicts given correct constants
S=4
modelTgivenL = function(L){
gtrue = 9.8*39.37 # inches/s^2
return(sqrt(2/gtrue)*(sqrt(L) - sqrt(L-S) ))
}
modelEstTgivenL = function(L){
constestimate = 0.0733 # from linear fit
return(constestimate*(sqrt(L) - sqrt(L-S) ))
}
png("drop.png",width=1024,height=1024)
plot(t ~ l,pch="x",xlab="L height",ylab="sensor time",cex=2)
points(seq(10,30), modelTgivenL(seq(10,30)),type="b", col="red",cex=2)
points(seq(10,30), modelEstTgivenL(seq(10,30)),type="b", col="green",cex=2)
title("drop times. black=measured red=model green=estimated model")
dev.off()
```
<img src="drop.png">
- RESULT: modeling and empirical data line up pretty well. Times are a
bit high for 20 inches (did I consistently hold it too low?)
- TODO: I plugged in the measured distance between sensors = 4
inches. Could I have estimated this also to get two estimates, g and
S, from the data? g can compensate S so you can't detangle.
================================
# Timer Construction
- For timer construction see:
## <a href="README_RESULT_arduino-break-beam.html">README_RESULT_arduino-break-beam.html</a>
]]>