# Modeling Center of Mass and Tipping Angle

## Materials:

• writing utensil
• calculator
• digital device with spreadsheet program
• digital device with internet access

## ObjectiveS

1. Apply a rigid-solid model to predict the tipping angle of objects on a slope.
2. Apply a uniform density model to calculate the center of mass of a human-shaped test object.
3. Experimentally test the tipping angle of the human shaped test object.
4. Compare the model prediction to the experimental results.
5. Apply a proportional static friction model to determine if the object will tip or slip.

## Modeling

### Tipping Angle

According to unit 5, weight, static friction, and normal force must be balanced in all directions in order for a body to remain in static equilibrium. The following video shows how to predict the maximum slope for which an object will not tip.

1) As see in the previous video, the maximum slope before tipping only depends on the center of mass height (h) and perpendicular distance (d) from the center of mass to the edge of the support base. Complete the following expressions according to the result shown in the video.

______, or in terms of the slope angle, _____.

### Tipping Angle

The following video demonstrates how to calculate the center of mass of an object with uniform density.

2) Calculate the center of mass height according to the method described in the video below. Record the result of your spreadsheet calculation for the center of mass height in the space below. Also record the perpendicular distance from center of mass to edge of support base, as discussed in the video.

3) Use your values above and the expression you completed in question (1) to predict the slope at which the object will tip. Show your work.

## Experimental methods

### Determining Tipping Angle

5) Watch the following video and record the maximum tipping angle.

7) Calculate the tangent of the tipping angle to get the slope.

8) Compare the predicted slope to the experimentally determined slope by calculating a percent difference. Cite any sources you used to find information on calculating percent difference.

## Conclusions

9)Do the experimental results provide evidence that the tipping model derived in the first video is valid under the conditions in this experiment? Explain.

## Further Questions

10) Assuming a uniform density model allowed us to gain experience with the concept of center of mass, but a uniform density model does not actually do well at predicting the center of mass of the human body. Explain why not and suggest a method for deterring the center of mass of a live person. Cite any and all sources used to answer this question.

Humans can stand on slopes much steep than would be predicted by the rigid object model demonstrated in the first video because the body is not rigid and we can shift our center of mass in the uphill direction as need to ensure it doesn’t move beyond the outermost edge of the downhill foot. These Ibex understand that concept well. Notice how they don’t let their center of mass move very far from the wall (don’t be fooled by the video title suggesting that the ibex is defying gravity, if it were then it wouldn’t really care where it’s center of mass was located!):

Notice that the ibex tend to slide when the wall is smooth and that they use tiny features in the wall to prevent sliding just as we used a thin metal ruler to prevent our model body from sliding. Without those features the slope would be limited by the static friction coefficient. In fact, the maximum slope before sliding has a very simple relation to the friction coefficient between the object and the slope as explained in the following video:

12) We want to apply the result found in the video above to calculate the maximum slope angle of unfeatured rock that a person could stand on without shoes.  To do that, we need a static friction coefficient between rock and skin. That depends significantly on the type of rock and also somewhat on the specific person’s skin. To get an estimate of the max slope, let’s assume a static friction coefficient between skin and rock average value of 0.45 reported by this study,[1] which tested 10 subjects at 6 different body parts for one particular type of rock.  Use the results of the video and the reported friction coefficient to estimate the maximum featureless slope for a shoeless human. Also calculate the angle of this slope. Show your work.

1. Zhang M, Mak AFT. In vivo friction properties of human skin. Prosthetics and Orthotics International. 1999;23(2):135-141. doi:10.3109/03093649909071625