torques

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Phy 201

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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Torques

Principles of Physics students need do only the first part.

General College Physics students need do only the first and second parts.

University Physics students should do all three parts.

The three parts are:

Rod supported by doubled rubber band, pulled down by two rubber bands

Simulating Forces and Torques on a Bridge

Torques Produced by Forces Not at Right Angles to the Rod

For this experiment you will use four of your calibrated rubber bands, a printed copy of the 1-cm grid (grid, a .gif file, or grid_1cm, a PDF), the threaded rod, 4 push pins and eight paper clips.

Rod supported by doubled rubber band, pulled down by two rubber bands

Setup

The setup is illustrated in the figure below. The large square represents the one-foot square piece of plywood, the black line represents the threaded rod, and there are six crude-looking hooks representing the hooks you will make by unbending and re-bending paper clips. The red lines indicate rubber bands. The board is lying flat on a tabletop. (If you don't have the threaded rod, you can use the 15-cm ramp in its place. Or you can simply use a pencil, preferably a new one because a longer object will give you better results than a short one. If you don't have the plywood and push pins, you can use the cardboard and 'staples' made from paper clips, as suggested in the Forces experiment.)

The top rubber band is attached by one hook to the top of the plywood square and by another hook to the approximate center of the rod. We will consider the top of the square to represent the upward direction, so that the rod is considered to be suspended from the top rubber band and its hook.

Two rubber bands pull down on the rod, to which they are attached by paper clips. These two rubber bands should be parallel to the vertical lines on your grid. The lower hooks are fixed by two push pins, which are not shown, but which stretch the rubber bands to appropriate lengths, as specified later.

The rubber band supporting the rod from the top of the square should in fact consist of 2 rubber bands with each rubber band stretched between the hooks (each rubber band is touching the top hook, as well as the bottom hook; the rubber bands aren't 'chained' together).

The rubber bands will be referred to by the labels indicated in the figure below. Between the two hooks at the top the rubber band pair stretched between these notes will be referred to as A; the rubber band near the left end of the threaded rod will be referred to as B; and the rubber band to the right of the center of the rod as C.

In your setup rubber band B should be located as close as possible to the left-hand end of the threaded rod. Rubber band C should be located approximately halfway, perhaps a little more, from the supporting hook near the center to the right-hand end of the rod. That is, the distance from B to A should be about double the distance from A to C.

Rubber band C should be stretched to the length at which it supported 10 dominoes (in the calibration experiment), while rubber band B should be adjusted so that the rod remains horizontal, parallel to the horizontal grid lines.

(If there isn't room on the plywood to achieve this setup:

First be sure that the longer dimension of the plywood is directed 'up-and-down' as opposed to 'right-and-left'.

Be sure you have two rubber bands stretched between those top hooks.

If that doesn't help, re-bend the paper clips to shorten your 'hooks'.

If the system still doesn't fit, then you can reduce the length to that required to support a smaller number of dominoes (e.g., 8 dominoes and if that doesn't work, 6 dominoes).

Data and Analysis: Mark points, determine forces and positions

Mark points indicating the two ends of each rubber band. Mark for each rubber band the point where its force is applied to the rod; this will be where the hook crosses the rod. Your points will be much like the points on the figure below. The vertical lines indicate the vertical direction of the forces, and the horizontal line represents the rod.

Disassemble the system, sketch the lines indicating the directions of the forces and the rod (as shown in the above figure). Make the measurements necessary to determine the length of each rubber band, and also measure the position on the rod at which each force is applied.

You can measure the position at which each force is applied with respect to any point on the rod. For example, you might measure positions from the left end of your horizontal line. In the above figure, for example, the B force might be applied at 3 cm from the left end of the line, the A force at 14 cm from the left end of the line, and the C force at 19 cm from the left end.

indicate the following:

In the first line, give the positions of the three points where the vertical lines intersect the horizontal line, in order from left to right.

In the second line give the lengths of the rubber band systems B, A and C, in that order.

In the third line give the forces, in Newtons, exerted by the rubber band systems, in the same order as before.

In the fourth line specify which point was used as reference point in reporting the three positions given in the first line. That is, those three positions were measured relative to some fixed reference point; what was the reference point?

Starting in the fifth line, explain how the forces, in Newtons, were obtained from your calibration graphs.

Beginning in the sixth line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> (note A doubled) intersections B A C, lengths B A C, forces B A C, reference point, how forces determined

******** ******** Your answer (start in the next line):

.6cm, 6.7cm, 11.8cm

6.8cm, 7.3cm, 6.7cm

1.52N, .38N, 1.52N

Left of the horizontal line

Use by the calibration graph.

The first is the intersection of horizontal line, lengths of the rubberbands, then the force exerted by each.

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Analyze results:

Vertical equilibrium: Determine whether the forces are in vertical equilibrium by adding the forces to obtain the net force, using + signs on upward forces and - signs on downward forces.

Give your result for the net force in the first line below.

In the second line, give your net force as a percent of the sum of the magnitudes of the forces of all three rubber band systems.

Beginning in the third line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> Fnet, Fnet % of sum(F)

******** Your answer (start in the next line):

3.42

1.52N-.38N+1.52N=2.66N

78%

#$&*

Rotational equilibrium: We will regard the position of the central supporting hook (the hook for system A) to be the fulcrum around which the rod tends to rotate. Determine the distance from this fulcrum to the point of application of the force from rubber band B. This distance is called the moment-arm of that force. Do the same for the rubber band at C.

report the moment-arm for the force exerted by the rubber band system B, then the moment-arm for the system C. Beginning in the second line, briefly explain what the numbers mean and how you obtained them.

----->>>>>>>>

******** Your answer (start in the next line):

11.4cm, 10.8cm

Measured from the fulcrum to b and c rubberbands.

#$&* moment arms for B, C

Make an accurate scale-model sketch of the forces acting on the rod, similar to the one below. Locate the points of application of your forces at the appropriate points on the rod. Use a scale of 4 cm to 1 Newton for your forces, and sketch the horizontal rod at its actual length.

Give in the first line the lengths in cm of the vectors representing the forces exerted by systems B, A and C, in that order, in comma-delimited format.

In the second line give the distances from the fulcrum to the points of application of the two 'downward' forces, giving the distance from the fulcrum to the point of application of force B then the distance from the fulcrum to the point of application of. force C in comma-delimited format, in the given order.

Beginning in the third line, briefly explain what the numbers mean and how you obtained them.

----->>>>>>>> (4 cm to 1 Newton scale) lengths of force vectors B, A, C, distances of B and C from fulcrum:

******** Your answer (start in the next line):

6.08cm, 1.52cm, 6.08cm

11.4cm, 10.8cm

The first is the forces for B,A, and C and the second is the distance from the fulrum to the bands.

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The force from rubber band C will tend to rotate the rod in a clockwise direction. This force is therefore considered to produce a clockwise torque, or 'turning force', on the rubber band. A clockwise torque is considered to be negative; the clockwise direction is considered to be the negative direction and the counterclockwise direction to be positive.

When the force is exerted in a direction perpendicular to the rod, as is the case here, the torque is equal in magnitude to the product of the moment-arm and the force.

What is the torque of the force exerted by rubber band C about the point of suspension, i.e., about the point we have chosen for our fulcrum?

Find the torque produced by rubber band B about the point of suspension.

Report your torques , giving the torque produced by rubber band B then the torque produced by the rubber band C, in that order. Be sure to indicate whether each is positive (+) or negative (-). Beginning in the next line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> torque C, torque B

******** Your answer (start in the next line):

17.3N*cm, -16.4 N*cm

I multipilied the force by the distance from fulcrum.

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Ideally the sum of the torques should be zero. Due to experimental uncertainties and to errors in measurement it is unlikely that your result will actually give you zero net torque.

Express the calculated net torque--i.e, the sum of the torques you have found--as a percent of the sum of the magnitudes of these torques.

Give your calculated net torque in the first line below, your net torque as a percent of the sum of the magnitudes in the second line, and explain starting at the third line how you obtained this result. Beginning in the fourth line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> tau_net, and as % of sum(tau)

******** Your answer (start in the next line):

.9

2.7%

I divided the net by the sum and got these numbers.

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Physics 121 students may stop here. Phy 121 students are not required to do the remaining two parts of this experiment, but may do so if they wish.

Simulating Forces and Torques on a Bridge

The figure below represents a bridge extended between supports at its ends, represented by the small triangles, and supporting two arbitrary weights at arbitrary positions (i.e., the weights could be anything, and they could be at any location).

The weights of the objects act downward, as indicated by the red vectors in the figure. The supports at the ends of the bridge hold the bridge up by exerting upward forces, represented by the upward blue vectors.

If the bridge is in equilibrium, then two conditions must hold:

1. The total of the two upward forces will have the same magnitude as the total of the two downward forces. This is the conditional of translational equilibrium. That is, the bridge has no acceleration in either the upward or the downward direction.

2. The bridge has no angular acceleration about any axis. Specifically it doesn't rotate about the left end, it doesn't rotate about the right end, and it doesn't rotate about either of the masses.

Setup

We simulate a bridge with the setup indicated below. As in Part I the system is set up with the plywood square, and with a 1-cm grid on top of the plywood.

The threaded rod will be supported (i.e., prevented from moving toward the bottom of the board) by two push pins, and two stretched rubber bands will apply forces analogous to the gravitational forces on two weights supported by the bridge.

Stretch one rubber band to the length at which it supported 8 dominoes in the calibration experiment, and call this rubber band B. Stretch the other to the length that supported 4 dominoes and call this rubber band C. Rubber band C should be twice as far from its end of the rod as rubber band B is from its end, approximately as shown below.

Use push pins (now shown) to fix the ends of the hooks and keep the rubber bands stretched.

Note that the length of the threaded rod might be greater than the width of the board, though this probably won't occur. If it does occur, it won't cause a serious problem--simply place the push pins as far as is easily feasible from the ends and allow a little overlap of the rod at both ends.

Be sure the rubber bands are both 'vertical'--running along the vertical lines of the grid. It should be clear that the push pins are each exerting a force toward the top of the board.

Place two more rubber bands, with the hooks at the positions of the push pins, as indicated below. Stretch these rubber bands out simultaneously until their combined forces and torques just barely begin to pull the rod away from the push pins supporting it. Fix push pins through the free-end hooks, so that the two new rubber bands support the rod just above the push pins supporting it, as close to the supporting pins as possible.

Remove the supporting pins. This should have no effect on the position of the rod, which should now be supported in its original position by the two new rubber bands.

Mark the ends of each of the four rubber bands, and also the position of the rod. Your marks should be sufficient to later construct the following picture:

Now pull down to increase the length of the rubber band C to the length at which that rubber band supported the weight of 10 dominoes, and use a push pin to fix its position.

This will cause the lengths of the rubber bands A, B and D to also change. The rod will now lie in a different position than before, probably at some nonzero angle with horizontal.

Mark the position of the rod and the positions of the ends of the four rubber bands, in a manner similar to that used in the previous picture. Be sure to distinguish these marks from those made before.

Analyze your results

The figure below indicates the first set of markings for the ends of the rubber bands, indicated by dots, and the line along which the force of each rubber band acts. The position of the rod is indicated by the horizontal line. The force lines intersect the rod at points A, B, C and D, indicated by x's on the rod.

From your markings determine, for the first setup, the length of each rubber band and, using the appropriate calibration graphs or functions, find the force in Newtons exerted by each.

Sketch a diagram, to scale, depicting the force vectors acting on the rod. Use a scale of 1 N = 4 cm. Label each force with its magnitude in Newtons, as indicated in the figure. Also label for each force the distance along the rod to its point of application, as measured relative to the position of the leftmost force.

In the figure shown here the leftmost force would be the 2.4 N force; its distance from itself is 0 and isn't labeled. The 5 cm, 15 cm and 23 cm distances of the other forces from the leftmost force are labeled.

For the first setup (before pulling down to increase the force at C), give the forces, their distances from equilibrium and their torques, in comma-delimited format with one torque to a line. Give lines in the order A, B, C and D. Be sure your torques are positive if counterclockwise, negative if clockwise. Beginning in the following line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> (ABCD left to right, position wrt A) four forces, four dist, four torques

******** Your answer (start in the next line):

0, 1.2cm, 10.1 cm, 13.9cm

1.50N, -1.6N, -.76N, 1.14N

0, -1.92N*cm, -7.68N*cm, 15.85N*cm

The first is the distance from the left, the second is the force exerted, then the torque is the force times the distance from the left.

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In the figure shown above the sum of all the vertical forces is 2.4 N + 2.0 N - 3.2 N - 1.6 N = 4.4 N - 4.8 N = -.4 N. Is this an accurate depiction of the forces that actually acted on the rod? Why or why not?

In the first line give the sum of all the vertical forces in your diagram. This is the resultant of all your forces.

In the second line, describe your picture and its meaning, and how well you think the picture depicts the actual system..

----->>>>>>>> (from scaled picture) sum of vert forces, describe picture and meaning

******** Your answer (start in the next line):

+1.50N+1.14N-1.6N-.76N=.28N

There are two forces upward and two forces downward both acting on the pole with the forces and distances labeled.

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In the figure shown above the 1.6 N force produces a clockwise torque about the leftmost force (about position A), a torque of 1.6 N * 15 cm = 24 N cm. Being clockwise this torque is -24 N cm. The 2.0 N force at 23 cm produces a clockwise torque of 2.0 N * 23 cm = 26 N cm. Being counterclockwise this torque is +26 N cm.

In the first line below give the net torque produced by the forces as shown in this figure. Beginning in the second line describe your picture and discuss whether it could be an accurate depiction of torques actually acting on a stationary rod. Support your discussion with reasons.

----->>>>>>>> net torque from given picture, describe your picture

******** Your answer (start in the next line):

0-1.92N*cm-1.68N*cm+15.85N*cm=6.25N*cm

It seems reasonable because both rubberbans are pulling back and forth to keep it stable.

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Now calculate your result

What is the sum for your diagram of the torques about the point of action of the leftmost force (i.e., about position A)? This is your experimentally observed resultant torque about A. Give your result in the first line below.

For your diagram what is the magnitude of your resultant force and what is the sum of the magnitudes of all the forces acting on the rod? Give these results in the second line in comma-delimited format.

Give the magnitude of your resultant force as a percent of the sum of the magnitudes of all the forces. Give this result in the third line.

For your diagram what is the magnitude of your resultant torque and what is the sum of the magnitudes of all the torques acting on the rod? Give these two results, and the magnitude of your resultant torque as a percent of the sum of the magnitudes of all the torques, as three numbers in your comma-delimited fourth line.

Beginning in the fifth line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> sum(tau) about A, Fnet and sum(F), Fnet % of sum(F), | tau_net |, sum | tau |, |tau_net| % of sum|tau|

******** Your answer (start in the next line):

25.45N*cm

.28N, 5N

5.6%

6.25N*cm

25.45N*cm

25%

I added the ones for the sum and for the net force or torque i used the neg or pos signs and added them then divided the resultant by the total.

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Perform a similar analysis for the second setup (in which you increased the pull at C) and give your results below:

For your diagram, what is the sum of the torques about the point of action of the leftmost force (i.e., about position A)? This is your experimentally observed resultant torque about A. Give your result in the first line below.

For your diagram what is the magnitude of your resultant force and what is the sum of the magnitudes of all the forces acting on the rod? Give these results in the second line in comma-delimited format.

Give the magnitude of your resultant force as a percent of the sum of the magnitudes of all the forces. Give this result in the third line.

For your diagram what is the magnitude of your resultant torque and what is the sum of the magnitudes of all the torques acting on the rod? Give these two results, and the magnitude of your resultant torque as a percent of the sum of the magnitudes of all the torques, as three numbers in your comma-delimited fourth line.

Beginning in the fifth line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> (pull at C incr) sum(tau) about A, Fnet and sum(F), Fnet % of sum(F), | tau_net |, sum | tau |, |tau_net| % of sum|tau|

******** Your answer (start in the next line):

42.96N*cm

.3N, 6.92N

4.3%

42.96N*cm, 4.58N*cm, 11%

First is the sum of torques, second is resultant force and net force, third is resultant force as a percent, fourth is sum of torques, resultant torque and it as a percent.

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For the second setup, the forces were clearly different, and the rod was not completely horizontal. The angles of the forces were therefore not all 90 degrees, though it is likely that they were all reasonably close to 90 degrees.

Look at your diagram for the second setup. You might want to quickly trace the lines of force and the line representing the rod onto a second sheet of paper so you can see clearly the directions of the forces relative to the rod.

In the first setup, the forces all acted in the vertical direction, while this may not be the case in this setup.

In the second setup, were the forces all parallel to one another? If not, by about how many degrees would you estimate they vary? Include a brief explanation of what your response means and how you made your estimates.

----->>>>>>>> (incr pull at C) variation of forces from parallel

******** Your answer (start in the next line):

5 to 10 degrees of variance.

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Estimate the angles made by the lines of force with the rod in the second setup, and give your angles in comma-delimited format in the first line below. Your angles will all likely be close to 90 degrees, but they probably won't all be 90 degrees. The easiest way to estimate is to estimate the deviation from 90 degrees; e.g., if you estimate a deviation of 5 degrees then you would report an angle of 85 degrees. Recall that you estimated angles in the rotation of a strap experiment.

Starting in the second line give a short statement indicating how you made your estimates and how accurate you think your estimates were.

----->>>>>>>> angles of lines of force with rod

******** Your answer (start in the next line):

95degrees, 85degrees, 80 degrees, 100 degrees

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Torques Produced by Forces Not at Right Angles to the Rod

Setup and Measurement

Set up a system as illustrated below.

As in our very first setup, the 'top' rubber band will in fact consist of two rubber bands in parallel.

The leftmost rubber band will remain vertical, while the rightmost rubber band will be oriented at a significant angle with vertical (at least 30 degrees).

The rightmost rubber band will be stretched to a length at which it supports the weight of 10 dominoes, and its point of attachment will be at least a few centimeters closer to that of the center rubber band than will the leftmost rubber band.

The leftmost rubber band will be stretched to the length at which it supports 8 dominoes.

Mark the ends of the rubber bands, the points at which the forces are exerted on the central axis of the rod, and the position of the central axis of the rod.

Measure the positions of the ends of the rubber bands:

Disassemble the system and draw an x and a y axis, with the origin somewhere below and to the left all of your marks.

Measure the positions of the ends of the rubber bands. Measure both the x and y coordinate of each of these positions, and measure each coordinate in centimeters.

Give in the first line below the x and y coordinates of the ends of the leftmost rubber band, which we will call rubber band system B. Give four numbers in comma-delimited format, the first being the x and y coordinates of the lower end, the second being the x and y coordinates of the upper end. All measurements should be in cm.

In the second line give the same information for the two-rubber-band system above the rod, which we will call system A.

In the third line give the same information for the rightmost rubber band which we will call system C.

Beginning in the fourth line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> (BAC) endpts of B, endpts of A, endpts of C

******** Your answer (start in the next line):

:

#$&*

Analysis

Using your coordinates and the Pythagorean Theorem, find the length of rubber band system B.

Do this by first finding the difference in the x coordinates of the ends of this band, then the difference in the y coordinates of the ends.

This gives you the lengths of the legs of a right triangle whose hypotenuse is equal to the length of the band.

Then using your calibration information find the force in Newtons exerted this system.

Do the same for systems A and C.

Give the length and force exerted by rubber band system B in the first line below, in comma-delimited format, then in the second and third lines give the same information for systems A and C. Starting in the fourth line give a brief description of what your results mean and how you obtained them.

----->>>>>>>> length and force of B, of A, of C

******** Your answer (start in the next line):

#$&*

Find the sine and the cosine of each angle with horizontal:

You earlier found the lengths of the x and y legs of the triangle whose hypotenuse was the length of rubber band system A.

The magnitude of the sine of the angle for the system the y component divided by the hypotenuse, i.e., the ratio of the y component to the hypotenuse. The sine is negative if the y component downward, positive if the y component is upward.

The magnitude of the cosine of the angle for the system the x component divided by the hypotenuse, i.e., the ratio of the x component to the hypotenuse. The cosine is negative if the x component is to the left, positive if the x component is to the right.

Find the sine and cosine for this system.

Using the same method find the sine and the cosine for system B and system C. Ideally system B will be acting vertically, so the cosine will be 0 and the sine will be 1; your measurements might or might not indicate a slight divergence from this ideal.

Report your results , giving in each line the sine and the cosine of the angle between the line of action of the force and the horizontal. Report lines in the order B, then A, then C. Beginning in the fourth line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> sin and cos of angle w horiz of B, A, C

******** Your answer (start in the next line):

#$&*

Find the angles of the force vectors with the horizontal, and the angles of the force vectors in the plane:

The angle of the force vector with horizontal is arcTan(y / x): the arctangent of the magnitude of the quantity you get with you divide the y component of the triangle used in the preceding, by the x component.

The arctangent is easily calculated using the 2d fn or inverse key on your calculator, along with the tan function.

The angle of the force vector in the plane is measured from the positive x axis, in the counterclockwise direction.

Give for each system the magnitude (i.e., the force in Newtons as you calculated it earlier), the angle with the x axis and the angle in the plane for each of the force vectors, reporting three comma-delimited lines in the order B, A and C. Starting in the fourth line briefly explain how you determined these values and how you obtained them:

----->>>>>>>> magnitude and angle of B, of A, of C

******** Your answer (start in the next line):

#$&*

Sketch a force diagram showing the forces acting on the rubber bands, using a scale of 1 N = 4 cm. Label the positions at which the forces act on the rod, the magnitude in Newtons of each force and the angle of each force as measured counterclockwise from the positive x axis (assume that the x axis is directed toward the right).

Find the components of each force:

Sketch the x and y components of each force vector, measure them and using the scale of your graph convert them back to forces. Then using the magnitude of the force and sine and cosine as found earlier, calculate each x and y component.

In the second line below you will report the x and y components of your sketch of vector A, the x and y components of the force of this system as calculated from the x and y components on your sketch, and the x and y components as calculated from the magnitude, sine and cosine. Report six numbers in this line, in comma-delimited format.

In the first line report the same information for vector B, and in the third line the same information for vector C.

Beginning in the fourth line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> comp of sketch, implied comp of force, comp calculated from mag and angle B, A, C

******** Your answer (start in the next line):

#$&*

Calculate the sum of the x components and of the y components, as determined by the magnitude, sine and cosine.

What is the sum of all your x components? What should be the sum of all the x components? How close is your sum to the ideal? Report as three numbers in comma-delimited format in line 1.

What is the sum of all your y components? What should be the sum of all the y components? How close is your sum to the ideal? Report as three numbers in comma-delimited format in line 2.

Beginning in the third line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> sum of your x comp, actual sum, how close to ideal x, then y

******** Your answer (start in the next line):

#$&*

The torque produced by a force acting on the rod is produced by only the component perpendicular to the rod. The component parallel to the rod has no rotational effect.

give in comma-delimited format a line for each force, indicating the distance of its point of action from that of the leftmost force, its component perpendicular to the rod, and its torque. The order of the lines should be B, A then C. Remember that torques should be reported as positive or negative.

Beginning in the fourth line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> (about B) dist from ref, perpendicular comp, torque for B, for A, for C

******** Your answer (start in the next line):

#$&*

Finally report the sum of your torques:

What is the sum of the torques about the point of action of the leftmost force? What should this sum be? How close is your sum to the ideal? Report as three numbers in comma-delimited format in line 1. Beginning in the second line, briefly explain what your results mean and how you obtained them.

----->>>>>>>> sum of torques, ideal sum, how close to ideal:

******** Your answer (start in the next line):

#$&*

Your instructor is trying to gauge the typical time spent by students on these experiments. Please answer the following question as accurately as you can, understanding that your answer will be used only for the stated purpose and has no bearing on your grades:

Approximately how long did it take you to complete this experiment?

******** Your answer (start in the next line):

#$&*

*#&!*#&!

@&

Your forces in the first setup didn't balance out. Clearly the rubber bands in the middle must have exerted more than .38 Newtons of force, but you're correctly reporting what the calibration graphs told you so that part is OK. Just be aware that the forces in fact do total zero, since the system was in equilibriurm.

Torques balance out nicely and your entire analysis looks good.

*@