Modeling Slide-Inducing Speed During Circular Motion

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

Materials

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

Objectives

1. 1. Apply Newton’s 2nd Law and an empirical static friction model to experimentally determine the static friction coefficient between two surfaces.
   2. Apply Newton’s 2nd Law, an empirical static friction model, and circular motion concepts to predict the rotation speed at which static friction fails to prevent a rotating object from sliding.
   3. Experimentally determine the rotation speed at which static friction fails to prevent a rotating object from sliding.
   4. Compare the model predictions with the experimental results to evaluate the quantitative static friction model.
   5. Compare the model predictions with the experimental results to evaluate and improve an incomplete interpretation of the static friction model.

Experimental methods

We will setup a small cylinder sitting on a rotating disk, with no other known forces beyond static friction acting horizontally on the object. We will then measure the speed at which the object begins to slide for several different values of the rotation radius and try to model the results. We will need to do a supplementary experiment to find the static friction coefficient ($\mu_s$) between the disk and the cylinder.

MOdeling

Max Rotation Speed

Fill in the blanks inside the brackets in the following model derivation.

An object moving in a uniform circle of radius $r$ must experience an acceleration of $a = v^2/r$. Applying Newton’s Second Law to an object moving in a uniform circle with mass $m$, we then have:

$F_{net} =[\,\,]v^2/r$

If static friction alone is providing the net force, then the net force is just static frictional.

$F_f =[\,\]v^2/r$

The max static friction force can be calculated from the empirical formula: $F^{max}_f  = \mu_s F_N$. Making that substitution allows us to determine the maximum tangential speed the object can have without slipping.

$ \mu_s F_N= [\,\,]v_{max}^2/r$

For the case of a flat disk, the normal force will be equal to the object’s weight. Near the surface of Earth we  calculate the weight as $F_g = mg$. Making that substitution we have:

$\mu_s[\,\,\,\ ]= [\,\,]v_{max}^2/r$

Cancel the masses and isolate $v_{max}^2$ to complete the following equation:

$v_{max}^2 =\,\,\,\,\,\,\,\,\,\,$

This max rational speed model depends on the static friction model: $F^{max}_f  = \mu_s F_N$. Therefore, data that support the validity of this model for $v_{max}$ also support the static friction model.

In order to use this model we will want to know the static friction coefficient. The following video shows a model for surface the angle at which an object begins to slide. Applying this model to experimental data will allow us to extract the static coefficient as a parameter.

Static Friction Coefficient from Slip Angle

Data Analysis

Static Friction Coefficient

1) Record the max angle before sliding from the video above.

2) Use the max angle before sliding to calculate the static friction coefficient ($\mu_s$).

Max Speed Data

3)  Create a spreadsheet with these columns:

4) Use the video at the start of the lab to record the times for two positions of the object just before it begins to slide for each radius trial. The times are seen on the stopwatch and the positions are indicated by the blue arrows. At the end of each trial in the video the blue arrows will remain in place so that you can determine the change in angle between the two positions.  Notice that the compass repeats every 90° so you will not be able to simply subtract one angle reading from the other, but instead you will need to take care to to determine the change in angular position. Record the times and change in angle in your spreadsheet.

Speed Data Analysis

5) Build formulas in your spreadsheet to calculate the remaining values in your chart for each radius.

6) Make a plot of $v_{max}^2$ vs. radius ($r$). Be sure to title your graph and label your axes, including units.

7) Apply a linear fit to your data and record the fit equation and R² value here:

8) If $v_{max}^2 = [\mu g] r$ and you plotted $v_{max}^2$ on the y-axis and  $r$ on the x-axis, then the slope of your fit should be __________.

9) Calculate a % difference between the slope you found from fitting the data and the slope calculated from the values above.

Conclusions

10) Our model for $v_{max}^2$ depends on the empirical quantitative static friction model ($F^{max}_f  = \mu_s F_N$). Does the data suggest support these two models? Explain. [Hint: Reference the results of the analysis section.]

Further Questions

Qualitative Static Friction Model

A common interpretation of the qualitative static friction model is that  “static friction is a responsive force that increases to be equal and opposite to whatever force is exerted, up to a maximum limit: $F^{max}_{f,s} = \mu_s F_N$. Once the applied force exceeds the maximum limit, the object slides”^[1]. 

However, our model assumes that static friction is the only horizontal force on the object, so our model does not agree with the interpretation above.  (According to the interpretation above, the friction force is “responsive” and can’t have a value unless some other force is applied to the object).

11) Adapt the previous interpretation to provide a more complete statement of the static friction model. (Provide an explanation of the responsive behavior of static friction on an object without the requirement that other forces to act on the object). Cite all sources. If desired, you may use the template below by filling in the in the blanks.

“The static friction force between two surfaces adjusts to whatever value is required to _________     __________ between the two surfaces, up to a maximum limit: $F^{max}_{f,s} = \mu_s F_N$. If the force required to _________     __________ is greater than the limit, then _________ occurs.”

Equivalence of Gravitational and Inertial Mass

12) Notice that while deriving the model we cancelled the inertial mass from Newton’s Second Law with the gravitational mass used to calculate the object’s weight. Therefore, testing this model also provided another test of the equivalence principle, similar to the test we did in the previous lab. Does this new experiment add evidence in favor or against the equivalence principle?

Inclusion of Tangential Acceleration

13) Notice that the disk and the cylinder were speeding up over time, so there was not only a centripetal acceleration pointing toward the center, but there was also a small tangential acceleration. The tangential can be found by calculating the change in speed divided by the change in time. For example, in the first trial the time to reach the slipping speed was roughly 60 seconds. Use that time and the slipping speed you found for that trial to calculate the tangential acceleration. Show your work.

The total acceleration is the vector sum of the centripetal and tangential accelerations. These accelerations are perpendicular so we can find the magnitude of the total acceleration with the Pythagorean Theorem:

$a = \sqrt{(\frac{v_{max}^2}{r})^2 + a_T^2}$.

Applying Newton’s Second Law with this total acceleration we get:

$\mu_s mg = m\sqrt{(\frac{v_{max}^2}{r})^2 + a_T^2}$

Cancelling the masses and solving for $v^2$ we find:

$v_{max}^2 = \sqrt{(\mu_s g)^2 – a_T^2}\cdot r$

Before including the tangential acceleration in our model we had: $v_{max}^2 = \mu_s g \cdot r$

14) In both cases the max speed is proportional to the radius, but the slope is different.  Calculating the percent difference between $\sqrt{(\mu_s g)^2 – a_T^2}$ and $\mu_s g$ allows us to estimate the percent error in the model caused by ignoring the the tangential acceleration. Do that with the tangential acceleration value you found for the first trial. Show your work.