The entire passage is written in the active voice.
Center of Mass of a Human
- Only students use the word 'human'. A title is the most important statement in the entire document. I would prefer
The location of the center
of gravity for males and females
Abstract
The purpose of this experiment is to locate the center of mass of people: to determine whether the center of mass is different for males and females, and to calculate the ratio of a person's center of mass to his/her height. - This is not an abstract. An abstract describes the research question and summarizes the findings. This statement is more like a scrambled research question.
Introduction
The center of mass is the balance point of an object's mass. - What is "the balance point of an objects mass"?
If a pivot were placed at this point, the object would remain in place and be balanced. - In a uniform gravitational field the center of mass and the center of gravity are in the same place, but they are not the same thing. The student has the right idea, but has insufficient control of the language to explain what they mean accurately. I would suggest.... If an object is suspended from the center of gravity, it will remain in equilibrium in any orientation.
The center of mass of a system is not always at the geometric center of the system. - True: but they mean center of gravity.
For example, a car's center of mass is closer to the ground rather than in the geometric center of the car so that the car is better balanced. - What exactly is meant by " so that the car is better balanced." If they mean that a car with a lower center of mass is more stable when cornering on a flat road they should say so.
Another example of this is the technique of a high jumper. A high jumper bends his body in a certain way so that the center of mass does not clear the bar, but the body does. - The expression is a little clumsy and the vague " ... in a certain way " could be made more specific with a diagram. I would prefer ... By bending his body over the bar, a high jumper is able to clear it, even thought it is above the maximum height of his center of mass.
Since the work to follow is about the average location of the center of gravity for males and females the introduction would be more relevant if the physical differences between males and females were discussed, rather than the position of the center of mass in a non uniform body (car) and the location of the center of gravity outside a bent body (high jumper). The two examples are not relevant to what follows. No research question arises from the points made in the introduction. One is supplied below.
Research question
Is the average ratio of the height of the center of mass to the height of the person (when standing to attention) the same for adolescent males and females?
This research question is now specific for body position and for the age of the subjects, which will be specifically defined in the data.
When a system is balanced around its center of gravity, it is said to be in a state of equilibrium. - I don't like the expression balanced around. Pivoted at or suspended from are better descriptions.
The center of mass can be referred to as a pivot point around which the system can revolve. - A system can be constrained to revolve about any point.
The system revolves due to the rotational equivalents of force, known as torques, which rotate the system either clockwise or counterclockwise. - No. Angular acceleration is induced by torque about a point - a revolving isolated system continues to revolve in the absence of torque. (Newton's first rule.)
Placing a pivot at the center of mass of a system results in that system being in equilibrium and having a net torque of zero. - Not necessarily - the net torque on a system has nothing to do with the location of a pivot. In equilibrium the net torque about any point is zero.
On each end of a long, rigid body, the torque on one end is equal in magnitude, but opposite in direction, to the torque on the other end, resulting in a net torque of zero. - Torque does not act at each the end of a body. Torque acts about a selected point.
The formula for torque is:
- No it isn't. The left hand side is a sum and the right hand side has one term only. The sigma is in the wrong place, and there are no italics.
Where r = radius and F = force. It is possible to locate the center of mass of a system by placing a pivot at the theoretical center of mass and using the formula for torque by setting the torques on either end of a long, rigid body equal to each other. - Oh dear. I think they mean something like this: When a body of length l is supported by two vertical forces, W1 and W2, (see figure 1), the sum of the moments about any point is zero. Taking moments about the center of gravity, a distance x from the right hand end, gives....
The relationship takes no account of the weight of the plank.
Show that the proper expression for x in terms of l, W1, W2, and W0 the weight of the plank that acts through the center of gravity at l/2 is ...
1 If W0 equals zero the expression simplifies to:
x = l [W2/(W1+W2)] 2 If W1= W2 the center of mass of the person is in the center of the uniform plank at l/2.
Fig 1 - a person lying on a plank supported by two force plates.
Procedure
1 Lay two 2 x 4 wooden beams of approximately 243 cm (8 ft) length side by side across two scales so that the ends of each beam are situated at the center of each scale. - not well defined end points.
2 Attach two meter sticks end to end on one side of one of the beams. This is done in order to measure one's height when a person is lying down. - syntax.
3 Record the weight of the beams without anyone on top of them. This is the tare weight.
4 Have a person remove his/her shoes, and then lay across the beams, making sure his/her heels line up with the end of one side of the beams. Have the person lay with their toes pointing upward and hands at their sides.
5 Record the weight on each scale. If the scales are not in the same units, use a conversion to make the data have common units, i.e. kilograms to pounds, pounds to kilograms.
6 Record the height of the person using the meter sticks that are attached to one of the wooden beams.
Note: this procedure section may be a description of a student-generated procedure but it appears to be a copied set of instructions. Note the slight but distinct improvement in expression and the rather bizarre statement in instruction 5.
Analysis
In order to determine the center of mass of a person, we used the formula for torque:
Note the zero phrase (In order to ... ) and the wrong formula.
The torques due to the weight on each end of the person were set equal to each other. - Torques, do not act on each end of the person as we discussed, but at least they are being consistent. They are taking moments.
The displayed weight on the scale was the force, and there are two different radii. The first radius is the distance to the center of mass from the person's feet, and the second radius is the length of the board minus the distance to the center of mass from the person's feet. OK
The net torque of the system is zero and therefore the torques on the opposite sides of the boards must be equal. - Oh dear.
where w1 is equal to the weight at the person's feet, x is equal to the distance from the person's feet to his/her center of mass, w2 is equal to the weight at the head, (left hand end of the plank) and l is equal to the length of the beams.
No italics but a correct equation - obtained by taking moments about the center of mass. Assuming that the weight of the plank is zero - which it is not, or that the center of mass of the body is at the mid point of the plank - which it is not.
The resulting formula, when solved for the distance to the center of mass from the person's feet (radius one) is:
After determining the location of each person's center of mass, the ratio of the center of mass to the height of each person was calculated using the formula:
where x is the location of the person's center of mass and h is the person's height. - to be strictly correct, x is the distance from the soles of the feet to the center of mass.
The original histograms were not to the same scale and they had no statistics with them. The replacements below are more instructive.
Males
Graph 1 - the frequency of the ratios of center of gravity to height for males.
Females
Graph 2 - the frequency of the ratios of center of gravity to height for males.
The original histograms were a good way of presenting the data for analysis but they did not have the same horizontal scale. The revision improves the presentation and while we are at it the mean and standard deviations and a comment about the deviation of the distribution from normal would be very helpful.
1 A person's center of mass is slightly below his/her belly button, which is nearly the geometric center of a person. - I don't see the location of the belly button anywhere in the data. The statement may be correct on both counts, but it cannot be made without either a reference, or data to support it.
2 Males and females have different centers of mass located in different places. - female centers of mass are relatively lower than those of males. - correct by inspection of the histograms, but how about giving a mean and standard deviation for both distributions - Physics is about numbers. The means of the two distribution differ by two standard deviations.
3 The average ratio of center of mass to height in females is approximately 0.543 ±? and the average ratio of center of mass to height in males is approximately 0.560 ±? - The use of approximately and not the more usual student use of about is good, but the statements are not finished off with an error estimate. There is in fact no meaningful discussion of errors and I doubt the significance of the third figure for two reasons. Biological variation means that the two deviations have means that differ by only two standard deviations and the individual data points are subject to errors due to uncertainty in the position of the ends of the plank, weight measurements, and weight of the plank which has been neglected.
Sources of Error
1 Clothing contributed to the weight of the subjects and therefore
resulted in a shift in each person's center of mass. -
Clothing might have had some effect, especially if it was unevenly
distributed. How could this quantified, and be overcome in future
measurements if found to be significant? The weight of the
planks would have had a greater effect. Why has this been ignored?
2 The subjects' heartbeats caused their center of mass to shift
because one's center of mass changes as the heart dilates and
constricts - pumping blood throughout the body. -
What a remarkable suggestion, and one that is easily tested.
Use force plates and plot the reaction forces as a function of
time. I expect the effect to be insignificant.
The raw data inclusions below are commendable, but the formatting and the use of significant figures sucks. The redundant figures have been deleted - the data could now be put in a proper table.
Female data
Vernier Format 2
Center of Mass of Females (revised).ga3 6/1/2006 20:02:04 .
Center of Mass of Females
Length of Board Weight at feet Weight at Head Height
of Person Center of Mass Ratio of Center of Mass to Height Average
Ratio of Center of Mass to Height
l W W h x x/h cm kg kg cm cm
243 34 19 164 87.1 0.531 0.543
243 29 15 156 82.8 0.531 0.543
243 34 20.5 166 91.4 0.549 0.543
243 34 21 169 92.78 0.549 0.543
243 28 14 149 81 0.542 0.543
243 33 18 164 85.7 0.523 0.543
243 41 23 157 87.3 0.556 0.543
243 41 25 165 92.0 0.558 0.543
243 39 22.5 166 88.9 0.536 0.543
243 32.5 19 161 89.6 0.557 0.543
243 38 22.5 165 90.4 0.548 0.543
243 27 15 160 86.8 0.542 0.543
243 36 21 163 89.5 0.549 0.543
243 37 22 173 90.6 0.524 0.543
243 34 19 162 87.1 0.538 0.543
243 36 22.5 168 93.5 0.556 0.543
Vernier Format 2
Center of Mass of Females (revised).ga3 6/1/2006 20:02:04 .
Histogram
Ratio of Center of Mass to Height - Bin Ratio of Center of Mass
to Height - Hist
x/h-Bin x/h-Hist
0.52 2
0.53 4
0.54 6
0.55 4
Male data
Vernier Format 2
Center of Mass of Males (revised).ga3 6/1/2006 20:00:06 .
Center of Mass of Males
Length of Board Weight at feet Weight at Head Height
of Person Center of Mass Ratio of Center of Mass to Height Average
Ratio of Center of Mass to Height
l W W h x x/h cm kg kg cm cm
243 40 26 170 95.7 0.563 0.56
243 39 29 179103.6 0.579 0.56
243 40 24 166 91.12 0.547 0.56
243 32 18.5 164 89.0 0.543 0.56
243 40 25 166 93.4 0.561 0.56
243 38 26 177 98.7 0.558 0.56
243 40 30.5 180 105.1 0.584 0.56
243 36 25 177 99.5 0.563 0.56
243 43 27 168 93.7 0.558 0.56
243 43 24.5 164 88.2 0.538 0.56
243 50 39 186 106.5 0.571 0.56
243 42 28 172 97.2 0.565 0.56
243 44 23 158 83.4 0.528 0.56
243 57 37 166 95.6 0.5766 0.56
243 53 36 173 98.3 0.568 0.56
243 36 22 166.5 92.2.556 0.56
Vernier Format 2
Center of Mass of Males (revised).ga3 6/1/2006 20:00:06 .
Histogram 2
Ratio of Center of Mass to Height - Bin Ratio of Center of Mass
to Height - Hist
x/h-Bin x/h-Hist
0.52 1
0.53 1
0.54 2
0.55 3
0.56 5
0.57 3
0.58 1