Challenge Experiment: Tracker andVertical Loops
Equipment:Video Capture, Tracker Analysis program, string, washers, rulers, plastic cups or lids, water
Challenge: Take a plastic cup full of water, hold it with a straight arm, and whirl it repeatedly in a vertical circle around your head. Try whirling the cup at different speeds. What do you expect to happen as you slow down the rotation speed. What determines whether you end up with wet or dry hair.
If you spin your arm slowly enough, thenthe water’s natural trajectory – a parabola – is below the circular arc and the water has time to fall out of the cup while the cup travels further along the circle. If you spin your arm quickly, the natural parabolic trajectory of the water is upwards above the circle, but the cup traps the water and pushes it along the circular arc so it can’t fall from the cup.
We don’t need to use a cup of water. A string with a mass on it can be used instead. However, a mass on a string rotates with Non-Uniform Circular Motion. The mass is moving at a higher speed at the bottom of the circle than at the top.
Let’s analyse this case of vertical motion. As the water rotates in the cup, it would like to follow a quadratic-type trajectory. Some combination of the acting forces is keeping the water rotating in a circle.
There are only two forces acting on the water … a downwards weight force W=mg and the inwardly directed force variously labelled the tension or normal force N. The only net force on the water is constructed from these two forces. In the diagram, there are four different tension forces, at the bottom Nb, at the right Nr, at the top Nt, and on the left NL.
Circular motion then requires that the centrally directed net force, or centripetal force Fcent, is given by Fcent = mv2/r.
Write down four relations at the bottom, right, top and left side of the circles giving the centrally directed force (noting there are different velocities at every point):
Bottom:mvb2/r = Fcent = Right:mvr2/r = Fcent =
Top:mvt2/r = Fcent = Left:mvL2/r = Fcent =
Rearrange these relations to give the tension in the string at various points in the circle:
Nb = Nr =
Nt = NL =
At what point in the circle is the tension the largest:
If the string is going to break, at what point in the rotation would it break:
At what point is the tension in the string the smallest:
If we keep our hand still and don’t put energy into the mass as it rotates, and if it loses a negligible amount of energy, then energy is conserved. The total energy Etot = KE + PE is always a constant. Write down the total energy at the bottom of the loop at height h=0, at the top of the loop at height h=2r, and at some arbitrary point in between at height h where the velocity is v:
Ebottom =
Etop =
Eh =
All of these energies are equal to each other. Let us say that they all equal a constant E. Notice that there is a linear relationship between v2 and h at any point. Write down this relationship
v2 =
If we plot y=v2 and x=h, we get a straight line y=mx+c. Write down a formula for your expected gradient and Y axis intercepts:
Gradient =
Intercept =
Rearrange your relation to make h the subject as a quadratic function of v in the form y=ax2+bx+c. Write a formula for your expected value of “a” and the intercept with the Y axis.
a =
Intercept =
Aim: To investigate the relationship between v and h for vertical circular motion, and to use this to measure the acceleration due to gravity.
Hypothesis: That v2 varies linearly with h, so a determination of the gradient allows a measurement of the acceleration due to gravity. And that h varies quadratically with v so quadratic regression gives a measurement of the acceleration due to gravity.
Data and Analysis:
Use as large a radius as possible to slow down the motion. Measure the radius of the String:
r =
The frame rate is probably 30frames per second (unless you put the camera into “SPORT” mode where the frame rate is doubled). This means that the time between images is probably 1/30s.
Swing a string of length r in a vertical loop. Video the loop and load the video into Tracker. Do the required data analysis, and then export the data from Tracker into Excel. You can choose to get Tracker to do all the work – Right Clicking on the graph in Tracker and choose to show velocity v and export v, h, and t into Excel. Alternatively, I export x,y, t into Excel and then use formulas to calculate v and h.
For two neighbouring points, the mass has moved from point (x1,y1) to point (x2,y2). Using Pythagorus, the distance moved is
d =
This movement occurred in a time of (t2-t1) and so the velocity was
v = d/(t2-t1) =
This velocity occurred at an average height h = (y2 + y1)/2.
Your data should look something like this.(Tracker is probably using cm as units, so don’t forget to correct this to m.)
t / x (cm) / y (cm) / x (m) / y (m) / v (m/s)0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 5.4222
0.03 / 17.80 / 2.70 / 0.18 / 0.03 / 5.3731
0.07 / 33.87 / 10.47 / 0.34 / 0.10 / 5.2295
0.10 / 46.63 / 22.25 / 0.47 / 0.22 / 5.0039
0.13 / 55.20 / 36.50 / 0.55 / 0.36 / 4.7167
0.17 / 59.41 / 51.60 / 0.59 / 0.52 / 4.3917
x (m) / y (m) / d (m) / v (m/s) / v^2 (m^2/s^2) / h (m)
0.00 / 0.00
0.18 / 0.03 / 0.18 / 5.40 / 29.18 / 0.01
0.34 / 0.10 / 0.18 / 5.35 / 28.66 / 0.07
0.47 / 0.22 / 0.17 / 5.21 / 27.16 / 0.16
0.55 / 0.36 / 0.17 / 4.99 / 24.88 / 0.29
Now you have to construct the velocity and height data (or simply export it from Tracker). Use formulas for distance d and v above.
Use your data to make a plot of v2 vs h.
Use the gradient to calculate g.
Gradient =
Acceleration due to gravity g =
Now make a plot of h vs v, and use your plot to calculate a measured value of g. The quadratic regression gives
a =
From this, calculate your value of the acceleration due to gravity g =
Conclusion: What do you conclude about circular motion and the acceleration due to gravity.
Answers:
E.g. Bottom Fcent = Nb-mg, Right Fcent = Nr, E.g. Top Nt=Fcent-mg, Bottom Nb=Fcent+mg, vt2=rg, Ebottom = mvb2/2, Etop = mvt2/2 +2mgr, Eh = mv2/2 + mgh, v2 = -2gh+2E/m, gradient = -2g, intercept=2E/m, h=(-1/2g)v2+2E/m, d=sqrt[(x2-x1)2+(y2-y1)2], v=d/t.
Theoretical Data
x (m) / y (m) / d (m) / v (m/s) / v^2 (m^2/s^2) / h (m)0.00 / 0.00
0.18 / 0.03 / 0.18 / 5.40 / 29.18 / 0.01
0.34 / 0.10 / 0.18 / 5.35 / 28.66 / 0.07
0.47 / 0.22 / 0.17 / 5.21 / 27.16 / 0.16
0.55 / 0.36 / 0.17 / 4.99 / 24.88 / 0.29
0.59 / 0.52 / 0.16 / 4.70 / 22.12 / 0.44
0.60 / 0.66 / 0.15 / 4.38 / 19.19 / 0.59
0.57 / 0.79 / 0.13 / 4.04 / 16.36 / 0.73
0.52 / 0.91 / 0.12 / 3.71 / 13.80 / 0.85
0.45 / 1.00 / 0.11 / 3.41 / 11.61 / 0.95
0.37 / 1.07 / 0.10 / 3.13 / 9.82 / 1.03
0.29 / 1.12 / 0.10 / 2.90 / 8.42 / 1.10
0.21 / 1.16 / 0.09 / 2.71 / 7.36 / 1.14
0.13 / 1.19 / 0.09 / 2.57 / 6.62 / 1.17
0.05 / 1.20 / 0.08 / 2.48 / 6.14 / 1.19
-0.03 / 1.20 / 0.08 / 2.43 / 5.91 / 1.20
-0.11 / 1.19 / 0.08 / 2.43 / 5.89 / 1.19
-0.19 / 1.17 / 0.08 / 2.47 / 6.09 / 1.18
Linear Gradient -2g = -20.1722/s2.
This rearranges to give acceleration
g = -20.172/-2 = 10.6m/s2.
Quadratic equation is
h = (-1/2g) v2 + E/(mg)
Quadratic regression gives
-0.0753=-1/(2g) s2/m
This rearranges to give
g = 6.64m/s/s.