<jupyter_start><jupyter_text>Uniform circular motion Become familiar with circular motionA particle moves in a circle at a constant speed (**uniform circular motion**). * For a given value of the rotational speed, such as for instance 60 **rpm** (revolutions per minute), we want to calculate:    * the rate of rotation, measured in **hertz**, i.e. the number of rotations per second,    * the period of rotation, measured in seconds, i.e. the amount of time needed to perform a complete round along the circular path.* We want to calculate also the **angular velocity**, measured either in    * degrees per second  or in      * radians per second.    ![](img/annotatedCyclette.jpg)* Given the value of the radius, suppose 0.165 m, we want to calculate the **linear speed**, measured in $m/s$* We want to calculate also the intensity of the **centripetal acceleration**, measured in $m/s^2$&nbsp; Discussion of the problem Rotational speedThe rotational speed of an engine in some cases is measured in revolutions per minute (rpm), as we can see for instance from the instrument panel of a car.![](img/smallPanel.jpg) Frequency or rotation rateIf we want to measure the **rotation rate** in cycles per second, or revolutions per second, considering that     1 min = 60 swe have    rotation rate (cycles/s) = [rotation rate (rpm)] / 60 &nbsp;It should be underlined that 1  cycle/s = 1 Hz&nbsp; PeriodThe **period** of the rotation,  **T**, can be obtained as the reciprocal of the rotation rate            T = 1 / f    &nbsp; Angular velocityFrom the rate of rotation we can evaluate the **angular velocity**, measured in degrees per second, if we consider that a complete cycle along the circumference corresponds to an angular displacement of 360 degrees. Thus,             f (cycle/s) &sdot; 360 (deg) = &omega; (deg/s)      Similarly, if we consider that a complete cycle around the circumference corresponds to an angle of 2 $\pi$ radians, it follows that            f (cycle/s) &sdot; 2 &pi; (rad) = &omega; (rad/s)    &nbsp; Linear speedThe linear speed of a particle moving on a circular path is related to the angular velocity, measured in **rad/s**, through the equation    v  = &omega; &sdot; r    where **r** is the value of the radius of the circumference. If **r** is measured in meters, from the equation we obtain the linear speed measured in m/s. Centripetal accelerationIf a particle moves at constant speed along a circular trajectory  (**uniform circular motion**), the acceleration vector is along the radius, and it is always directed towards the center of the circle. The magnitude of the centripetal acceleration can be calculated as     ac = v2 / r and also as     ac = &omega;2 &sdot; r   Calculations with PythonIn the following Python code we compute the values of several variables related to the uniform circular motion and we show the results as Markdown text, or as a Markdown table, or as a HTML table.<jupyter_code>import numpy as np
from IPython.display import display, Markdown

def printmd(string):
    display(Markdown(string))

    
speed_rpm = 60        # rotation speed (rpm)

r = 0.165             # radius  r  (m)

f = speed_rpm / 60        # frequency (Hz)

T = 1/f                   # period (s)

omega = 2 * np.pi * f     # angular speed (rad/s)

omega_deg = 360 * f       # angular speed in  degrees/second

v = omega * r             # linear speed

a = omega * omega * r     # acceleration

print()

# display the results as Markdown text

printmd('#### Results as Markdown text')
printmd('<br> Rotation speed <br> *f*<sub>rpm</sub> = '+str(speed_rpm)+ ' rpm')
printmd('Radius <br> *r* =  '+str(r)+ ' m')
printmd('Frequency <br> *f* =  '+str(f)+ ' Hz')
printmd('Period <br>  *T* = '+str(T)+ ' s')
printmd('Angular speed <br> &omega; = '+str(omega)+ ' rad/s')
printmd('Angular speed <br> &omega;<sub>deg</sub> = ' + str(omega_deg) + ' deg/s')
printmd('Linear speed <br> v = ' + str(v) + ' m/s')
printmd('Acceleration <br> a<sub>c</sub> = '+ str(a) + ' m/s<sup>2</sup>')

print()

# display the results as a Markdown table

printmd('#### Results as a Markdown table')
printmd(
"""
|variable|symbol|value|unit|
|-------|:-:|-----|:-:|
|Rotation speed| f<sub>rpm</sub> | {one}|rpm |
|Radius| r | {two} | m|
| Frequency | f | {three:0.3f} | Hz |
| Period | T | {four:0.3f} | s |
| Angular speed | &omega;| {five:0.3f} | rad/s |
| Angular speed | &omega;<sub>deg</sub> | {six:0.3f} | deg/s |
| Linear speed | v | {seven:0.3f} | m/s |
| Acceleration | a<sub>c</sub> | {eight:0.3f} |  m/s<sup>2</sup> | 
""".format(one = speed_rpm, two = r, three = f, four = T, five = omega, six = omega_deg, seven = v, eight = a))

print()

# display the results as an HTML table

printmd('#### Results as an HTML table')

myResults = '<table style="border:1px solid green;margin-left:auto;margin-right:auto;padding:2px;border-spacing:20px;" >' + \
'<tbody><tr><td>Rotation speed</td><td>f<sub>rpm</sub></td><td>' + str(speed_rpm) + '</td><td>' + ' rpm'+'</td>' + \
'</tr><tr><td>'+ 'Radius'+ '</td><td>r</td><td>' + str(r)+ '</td><td>' + 'm' + '</td></tr>' + \
'<td>Frequency</td><td>f</td><td>'+str(format(f))+'</td><td>' + 'Hz' + '</td></tr>' + \
'<td>Period</td><td>T</td><td>'+str(T)+'</td><td>' + 's' + '</td></tr>' + \
'<td>Angular speed</td><td>&omega;</td><td>'+str(omega)+'</td><td>' + 'rad/s' + '</td></tr>' + \
'<td>Angular speed</td><td>&omega;<sub>deg</sub></td><td>'+str(omega_deg)+'</td><td>' + 'deg/s' + '</td></tr>' + \
'<td>Linear speed</td><td>v</td><td>'+str(v)+'</td><td>' + 'm/s' + '</td></tr>' + \
'<td>Acceleration</td><td>a<sub>c</sub></td><td>'+str(a)+'</td><td>' + 'm/s<sup>2</sup>' + '</td></tr>' + \
'</tbody></table>'
printmd(myResults)

print()<jupyter_output><jupyter_text>&nbsp;> If you want to run the Python code in the **binder** environment online,[![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/POSS-UniMe/simple-physics-with-Python/HEAD)&nbsp; Get a feel ofA particle performs a uniform circular motion on a circumference whose radius is $r = 2 m$.* What is the value of the rotational rate needed to produce a centripetal acceleration of $9.8 m/s^2$ (i.e. equal to the acceleration of gravity)?* What is the corresponding value of the linear speed?&nbsp;From $a_c = \omega^2 r$ we obtain$$ \omega = \sqrt{\dfrac{a_c}{r}} $$On the other hand, since $f = \dfrac{\omega}{2\pi}$ we have$$ f = \dfrac {1}{2\pi} \sqrt{\dfrac{a_c}{r}} $$and finally, $$ f_{rpm} = 60 \dfrac {1}{2\pi} \sqrt{\dfrac{a_c}{r}} $$&nbsp;The linear speed is$$ v = \omega r = r \sqrt{\dfrac{a_c}{r}} =  \sqrt{a_c r}$$  Python scriptWith the following script we make calculations and show the results in as a nicely formatted output.<jupyter_code>a = 9.8   # acceleration (m/s^2)
r = 2     # radius (m)
f_rpm = 30*np.sqrt(a/r)/np.pi
v = np.sqrt(a*r)
print()
display(Markdown(
    'Required **rotational speed** *f*<sub>rpm</sub> = {rotspeed:0.3f} rpm'.format(rotspeed=f_rpm)
))
display(Markdown(
    '**Linear speed** *v* = {linearspeed:0.3f} m/s = {speedkmh:0.3f} km/h'.format(linearspeed=v, speedkmh=v*3.6)
))
print()<jupyter_output>
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