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%matplotlib inline



# Modeling Static Friction

### Introduction:

From everyday experience, we know that it takes a certain amount of force to overcome friction and get an object sliding. The goal of this investigation is to model the dependence of this maximum static frictional force on the mass of the object in question.

### Procedure:

We attached a spring scale to the side of a hollow box on a level surface. Adding additional masses, we measured the amount of force it took to get the box to begin to slide. we did he same thing again but on an incline of 14 degrees.

### Data and Analysis:

$m_{added}$ $f_{flat}$ $f_{incline}$
0 0.6 0.76
100 1.1 1.39
200 1.8 1.93
300 2.3 2.28
400 2.6 2.75
500 3.1 3.11


In [29]:

import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import curve_fit

m = np.array([0,100,200,300,400,500,])
f = np.array([0.6,1.1,1.8,2.3,2.6,3.1])
i = np.array([0.76,1.39,1.93,2.28,2.75,3.11])
y = np.array([0, 0.95, 1.90, 2.85, 3.80, 4.75])
h = np.array([0, .98, 1.96, 2.94, 3.92,4.9])

mm = np.linspace(0, 600, 800)
nn = np.linspace(0, 5, 10)

def linear(m, a, b):
return a*m + b

a, b = curve_fit(linear, m, f)[0]
c, d = curve_fit(linear, m, i)[0]
g, h = curve_fit(linear, y, i)[0]

force = linear(mm, a, b)
incline = linear(mm, c, d)
normal = linear(mm, g, h)
print(a, b)
print(c, d)
print(g, h)

#plt.xlim(0,800)
plt.ylim(0,5)
plt.title("Modeling Static Friction")
plt.ylabel("Max Frictional Force")
plt.plot(mm, linear(mm, a, b), 'c--')
plt.plot(mm, linear(mm, c, d), 'r--')
plt.plot(m, f, 'b.')
plt.plot(m, i, 'g.')
plt.show()

plt.xlim(0,5)
plt.ylim(0,5)
plt.title("Modeling Static Friction")
plt.xlabel("Normal Force(N)")
plt.ylabel("Frictional Force(N)")
plt.plot(nn, linear(nn, g, h), 'k--')
plt.plot(y, i, 'ko')
plt.show()




0.00499999999783 0.666666666666
0.00462285714646 0.880952380135
0.486616541649 0.880952381523



## Conclusion

Our expiriment was to test if static friction fits a linear model. We first started by getting a box and testing with different masses how much force was needed to get it to move. We then use the force from the spring scale to measure the frictional force. To do this we put the force and mass into the formula $f_s = F_{sp} - mgsin\theta$, and theta being 14$^{\circ}$. We then did the same expiriment but on an incline at 14$^{\circ}$. We started by with an empty box and finding what amount of force was needed to get it to move or its static friction. The equation for static friction on a flat surface is $y = 0.005x + 0.667$ in black. Also, the equation of the static friction on an incline is $y = 0.005x + 0.881$ in red. The data tells us that the box on an incline will need more force to move it forward or up the incline than the force needed to pull the box on a flat surface with no incline. We did this expiriment to test if static friction followed a linear model or a quadratic model, and after the expiriment,we learned that we need more force to move a mass up a incline than on leveled ground.



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