We did a lab this week that involved the rotation of a meter stick about a
pivot. Using torque produced by the weight at the center of mass and the
rotational inertia about the pivot, students calculated a theoretical value
for angular velocity of the meter stick when it passed through the horizontal
after being released from a vertical position. They then went on to confirm
this experimentally.
This started me thinking about a solution for angle as a function of time for
this experiment, net torque = rotational inertia times angular acceleration.
This gives:
Torque = I(alpha)
alpha = Torque/I
[d^2 theta]/[dt^2] = Torque/I
Choosing x as the horizontal axis, y as the vertical axis, and measuring
theta conventionally, the torque provided by the weight is a function of
cos(theta) and the last equation above can be written:
[d^2 theta]/[dt^2] = A*cos(theta)
Where A depends on the rotating object's rotational inertia and the torque
provided by the weight.
As an example, a meter stick, with mass, m, and l = 1m, rotating about an end
has:
I = (ml^2)/3
Torque = -mglcos(theta)/2
A = -3g/(2l)
I proceeded numerically and then tested the solution in a spreadsheet.
Calculations are shown below where t is time, q is theta, w is omega, and a
is alpha (for lack of Greek symbols). The spreadsheet follows this analysis.
t q w a
--------------------------
0 qo wo ao
t q1 w1 a1
2t q2 w2 a2
3t q3 w3 a3
4t q4 w4 a4
. . . .
. . . .
Using finite element differentiation:
w1 = (q2-qo)/2t
w2 = (q3-q1)/2t
w3 = (q4-q2)/2t
a2 = (w3-w1)/2t
a2 = (q4-2q2+qo)/4t^2
Acos(q2) = (q4-2q2+qo)/4t^2
q4 = 4t^2Acos(q2)+2q2-qo (q4 depends on q2 and qo)
Similarly:
q5 = 4t^2Acos(q3)+2q3-q1 (q5 depends on q3 and q1)
Therefore, if qo through q3 can be generated, then qi for i>3 can be
generated and wi and ai can be calculated using finite element
differentiation.
I chose dt small and used a linear approximation for the first four rows in
the table. Then choosing initial values for dt, qo and wo:
The spreadsheet is shown below where A is for a meter stick rotating about
one end. Graphing q, w, and a versus time on a single graph is interesting.
Try copying the information below into cell A1 of a spreadsheet. You may
have to delete single quotes in front of formulas. If this doesn't work,
then create the spreadsheet from scratch. Fill the last row down a few
thousand times. Delete the last entry for omega and the last two entries for
alpha.