torques

Your 'torques' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

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Positions of the three points of application, lengths of systems B, A and C (left to right), the forces in Newtons exerted by those systems, description of the reference point:

0, 7.0, 14.0

7.9, 7.6, 7.6

.209, .201, .201

I used the left end of the rod as my reference point.

I multiplied each length by .0264N/cm to get the forces, as derived from the calibration experiment.

These calculations demonstrate what force is associated with rubber band systems B A C.

Net force and net force as a percent of the sum of the magnitudes of all forces:

.209N

34%

.201-.201+.209=.209. .201+.201+.209=.611. .209/.611

Moment arms for rubber band systems B and C

7.0cm, 7.0cm

These are the distances of B and C from the fulcrum.

Lengths in cm of force vectors in 4 cm to 1 N scale drawing, distances from the fulcrum to points B and C.

.836cm, .804cm, .804cm

7.0cm, 7.0cm

The first line is the force of B A C multiplied by 4cm/N. The second line is the distance of B C from the fulcrum.

Torque produced by B, torque produced by C:

1.46N cm, -1.41N cm.

.209N*7.0cm=1.46N cm. .201N*7.0cm=1.41N cm. These are the torques exerted by B and C on either side of the fulcrum of the rod.

Net torque, net torque as percent of the sum of the magnitudes of the torques:

.05N cm

1.46-.141=.05/(1.46+1.41)=2%

Forces, distances from equilibrium and torques exerted by A, B, C, D:

.206N, 0cm, 0N cm

.209N, 2cm, -.417N cm

.198N, 10cm, -1.98N cm

.195N, 14cm, 2.74N cm

Each force was rubber band stretch*.0264N/cm, from my calibration experiment. Each distance was from the leftmost force. Each torque was distance*force.

The sum of the vertical forces on the rod, and your discussion of the extent to which your picture fails to accurately describe the forces:

-0.2N.

My picture is similar to yours and likewise fails to completely balance vertical forces due to experimental error, although the net force is small.

Net torque for given picture; your discussion of whether this figure could be accurate for a stationary rod:

0.343N cm

Due to experimental error, my torques did not cancel, although I expect that the net torque is in fact 0 because the rod is stationary.

For first setup: Sum of torques for your setup; magnitude of resultant and sum of magnitudes of forces; magnitude of resultant as percent of sum of magnitudes of forces; magnitude of resultant torque, sum of magnitudes of torques, magnitude of resultant torque as percent of the sum of the magnitudes:

0.343N cm

-0.2N, 30.6N

.6%

0.343N cm, 4.347, 8%

This box shows my figures for resultant force as a percent of all forces and resultant torque as a percent of all torques for the first setup.

For second setup: Sum of torques for your setup; magnitude of resultant and sum of magnitudes of forces; magnitude of resultant as percent of sum of magnitudes of forces; magnitude of resultant torque, sum of magnitudes of torques, magnitude of resultant torque as percent of the sum of the magnitudes:

.116N cm

-0.037N, .835N

.116N cm, 5.28N cm, 2%

This box shows my figures for resultant force as a percent of all forces and resultant torque as a percent of all torques for the 2nd setup.

In the second setup, were the forces all parallel to one another?

In the 2nd setup, the forces remained very close to parallel, but some tilt was observed, mostly in the leftmost rubber band. I estimate its total tilt was less than 5 degrees. Considering that the leftmost rubber band is the fulcrum of the system and does not factor into torque calculations, this finding is not of great importance.

Estimated angles of the four forces; short discussion of accuracy of estimates.

85, 87, 90, 87

I think my estimates are accurate to within a couple degrees considering the width of the rubber band in relation to the tick marks on my protractor.

x and y coordinates of both ends of each rubber band, in cm

6.5, 4.2, 6.5, 11.5

7.2, 20.0, 14.7, 26.7

25.9, 4.0, 23.1, 11.8

These are the coordinates of the lower and upper limits of rubber bands B,A,C with the lower limit of rubber band B set to 0,0.

Lengths and forces exerted systems B, A and C:.

7.3cm,.193N

10.1cm, .266N

8.2cm, .216N

I used my coordinates and the Pythagorean Theorem to find the length of rubber band systems B A C and I multiplied each length by .0264 (as derived above) to find the associated force.

Sines and cosines of systems B, A and C:

1, 0

.772, .663

.951, .341

These are the sines and cosines for B A C found by dividing the vertical and horizontal components, respectively, by the hypotenuse of the system.

Magnitude, angle with horizontal and angle in the plane for each force:

.193N, 0, 0

.266N, 41, 131

.216N, 70, 340

First I copied the forces for B A C above. Then, using the technique above, I calculated the angles of the force vectors with the horizontal, and the angles of the force vectors in the plane.

x and y components of sketch, x and y components of force from sketch components, x and y components from magnitude, sine and cosine (lines in order B, A, C):

0cm, 7.3cm, 0N, .193N, 0, .193N

-4.2cm, 12.1cm, -.111N, .319N, -.205N, .176N

7.6cm, -15.3cm, .201N, -.404N, .205N, -.074N

For systems B A C, these are the vector length x and y components, the vector force x and y components calculated from my sketch, and the vector force x and y components calculated from the sine and cosine values obtained above.

Sum of x components, ideal sum, how close are you to the ideal; then the same for y components.

0, 0, 0

.102, 0, +.102

The sums should equal zero if the systems are in equilibrium, but there is some variance in the y component sum due to experimental error.

Distance of the point of action from that of the leftmost force, component perpendicular to the rod, and torque for each force:

0, .193N, 0N cm

10.0, .176N, 1.76N cm

12.0, .074, -0.888N cm

I calculated each torque as the force perpendicular to the rod times the distance from the leftmost point of action.

Sum of torques, ideal sum, how close are you to the ideal.

.872N cm, 0N cm, +0.872

Because the rod is stationary, there should be a zero net torque. The net positive torque indicates some experimental error.

How long did it take you to complete this experiment?

12hrs