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Spreadsheet Analysis of Rotating Stick



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:

ao = Acos(q0)
q1 = qo+wo(dt)
w1 = wo+ao(dt)
a1 = Acos(q1)

q2 = q1+w1(dt)
w2 = w1+a1(dt)
a2 = Acos(q2)

q3 = q2+w2(dt)
w3 = w2+a2(dt)
a3 = Acos(q3)

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.


dt 0.002
A -14.7
4t^2 =4*b1^2
4t^2A =b2*b3
2*dt =2*b1
theta initial 1.54
omega initial 0

t theta omega alpha
0 =b6 =b7 =$b$2*cos(b10)
=$b$1+a10 =b10+c10*$b$1 =c10+d10*$b$1 =$b$2*cos(b11)
=$b$1+a11 =b11+c11*$b$1 =c11+d11*$b$1 =$b$2*cos(b12)
=$b$1+a12 =b12+c12*$b$1 =c12+d12*$b$1 =$b$2*cos(b13)
=$b$1+a13 =$b$4*cos(b12)+2*b12-b10 =(b15-b13)/$b$5 =(c15-c13)/$b$5


Bob Carlson