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.
** Your optional message or comment: **
** 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, 7.45, 11.02
8.0, 7.9, 8.1
-.36, 1.94, -1.38
The point where rubber band B's hook touches the rod is the reference point
The forces were calculated using the regression equations from the best fit lines found during the rubber band calibration.
The results are the force in Newtons that each rubber band exerts on the rod.
** Net force and net force as a percent of the sum of the magnitudes of all forces: **
.207
5.6
The sum of the forces is .207. This represents a net upward force of .207 Newtons. A system in equilibrium should have a net force of zero. This result is probably due to a combination of uncertainty of measurement and friction. The net force divided by the sum of the magnitudes of the forces of the rubber bands is .056. The net force is 5.6 percent of the total force.
** Moment arms for rubber band systems B and C **
7.45, 3.57
The moment-arms of the forces, measured in cm. These are the distances between the forces and the fulcrum about which they act.
** Lengths in cm of force vectors in 4 cm to 1 N scale drawing, distances from the fulcrum to points B and C. **
1.44, 7.778, 5.52
7.45, 3.57
The first line lists the lengths of the vectors representing the forces of the rubber bands, using a scale of 4 cm to 1 Newton. The magnitude of the force in Newtons was multiplied by four. The second line lists the distances from each of the downward forces to the fulcrum.
** Torque produced by B, torque produced by C: **
-2.68, +4.912
These are the torques, in Newton-centimeters, that the rubber bands exert on the rod. Each is the product of the force and the distance from the force to the fulcrum. The positive torques tend to rotate an object in a counter-clockwise direction.
** Net torque, net torque as percent of the sum of the magnitudes of the torques: **
2.23
30
The sum of the torques is the net torque. The sum of the magnitudes of the torques is 7.59. The net force divided by this sum gives 30 percent. The net torque should be zero for a system in equilibrium. The error found in calculating the net force is being multiplied during the net torque calculation.
** Forces, distances from equilibrium and torques exerted by A, B, C, D: **
1.64, 0, 0
1.59, 1.3, -2.07
.895, 10.6, -9.49
.57, 14.1, 8.04
The forces in Newtons are calculated using the lengths of the rubber bands, and the corresponding regression equations found during calibration. The distances from the leftmost force are reported in cm. The torque about the point of application of the leftmost force is the product of the force and its distance from the leftmost 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: **
-.275
My picture has vectors representing the forces exerted by each of the rubber bands, at the distances they were exerted from the leftmost force. The leftmost force is also included. The sum of the vertical forces should be zero, so the net downward force of .275 Newtons is an error. This error is probably the result of uncertainty in measurement, and calibration of the rubber bands. Disassembling the prior setup, I noticed that there was some error due to friction holding the rod in place. This is not a likely source of error in this setup.
** Net torque for given picture; your discussion of whether this figure could be accurate for a stationary rod: **
+6
The figure shown cannot be an accurate depiction of the torques acting on a stationary rod, because the forces depicted tend to rotate the rod in a clockwise direction.
** 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: **
-4.15
-.275, 4.695
5.6
4.15, 19.06, 21
The sum of the torques about the point of action of the leftmost force is the sum of each product force * distance relative to the leftmost force. The result of -4.15 N * cm corresponds to a tendency to rotate clockwise. For a motionless system, this is not correct. The net force of -.275 N is also an error for a system not in motion. The two errors are consistent; a downward net force to the right of the reference would tend to rotate the system in a clockwise direction.
** 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: **
-.05
.286, 5.854
4.9
.5, 43.364, 1.1
The resultant torque is the sum of all the torques calculated for this setup. It is reported in N * cm. In this case, the error in net torque and net force is not consistent. An upward net force to the left of the reference would tend to rotate the rod counterclockwise.
** In the second setup, were the forces all parallel to one another? **
The rod is no longer horizontal, but angles downward to the right. The forces still appear to be parallel to each other. Their points of attachment to the rod and the hooks remain the same, and the change in position of the rod is very small.
** Estimated angles of the four forces; short discussion of accuracy of estimates. **
88, 90, 92, 88
The rod appears to rotate approximately 2 degrees. The angles between forces A, C, and D appear to be slightly changed from 90 degrees, but the angle between force B and the rod seems unchanged. I expected more change in the angles of the forces to show greater change, so I think my estimates may be incorrect.
** x and y coordinates of both ends of each rubber band, in cm **
3.82, 1.10, 3.82, 9.33
8.32, 11.67, 6.71, 19.64
17.95, 2.20, 13.99, 9.46
These are the endpoints of the rubber bands, measured in cm from a reference point below and to the left of the system.
** Lengths and forces exerted systems B, A and C:. **
8.23, .8
8.13, 2.63
8.27, 1.69
These are the lengths of the rubber bands connected to the rods, and the corresponding forces in Newtons. I used the regression equation from the calibration exercise to find the force exerted at each length. For the rubber bands attached to A, the force is the sum of the forces each rubber band exerts at the length these bands were stretched.
** Sines and cosines of systems B, A and C: **
1,0
-.98, .20
1.14, .48
These are the sin and cos of the angle each force forms with the rod. They were found by using the sin = opp/hyp and cos = adj / hyp identities and the measurements taken earlier. The x-components were considered negative if the slope of the line representing the force was negative, as for force A. The y-components were considered negative for forces B and C, since they are below the rod
** Magnitude, angle with horizontal and angle in the plane for each force: **
.80, 270
2.63, 101.4
1.69, 311.2
These are the forces exerted by the rubber bands, and the angle at which they exert them on the rod. The angles were found using the arctan function, with the exception of the first, vertical force (divide by zero error). The angle of the first force, with a vertical orientation, can be determined by inspection or by the arcsin function. For the force exerted at A, 180 degrees was added to orient the force in the correct quadrant. For the force at C, 360 degrees was added to orient it in the correct quadrant.
** 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): **
-.52, 2.58
0, 0.8
1.12, -1.26
These are the x- and y- components of each of the forces, A,B,C exerted on the rod. They were calculated as the product of the magnitude of the force and the cos (for x) or sin (for y) of the angle the force forms with the x-axis.
** Sum of x components, ideal sum, how close are you to the ideal; then the same for y components. **
.6, 0, 36%
.52, 0, 11%
These are the net forces in the x and y directions. Since the system is not in motion, the net forces in each direction sum to zero. The calculated forces have errors of 36 and 11 percent of the total forces exerted. This percentage is the result of dividing the net force by the sum of the magnitudes of the forces.
** Distance of the point of action from that of the leftmost force, component perpendicular to the rod, and torque for each force: **
0, .8, 0
4.5, 2.58, 11.61
10.17, -1.26, -12.81
The torques are the products of the magnitude of the y-component of the force and its distance from the point of reference. They relate to the tendency of the rod to rotate about the point of reference.
** Sum of torques, ideal sum, how close are you to the ideal. **
-1.20, 0, 5%
Since the system is not in motion, the net torque is zero. The calculated net torque has an error of 5 percent of the total torques. This percentage is the result of dividing the net torque by the sum of the magnitudes of the torques.
** How long did it take you to complete this experiment? **
About 8 hours, in three sittings
** Optional additional comments and/or questions: **
&#Your work looks good. Let me know if you have any questions. &#