Physics Toys, Inc. Tension Analysis: Unit III - Dynamics: AP Physics B Web Portfolio

Physics Toys, Inc. Tension Analysis

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Assumptions

Plane flies in a circular path
Plane flies at the same height around its height
Tension on string remains constant
Velocity of plane is constant
Length of string does not change while rotating

Procedure

First, take off the plane from the string to which it's attached. Measure the mass of the plane using a balance and record in data table. Then, reattach the plane to the string. Now, measure the length of the string, from the point at which the string is attached to the plane, to the point on the ceiling to which the string is attached. Record the length in data table. Turn on the plane's propeller. Start to fly the plane in a circle. Begin timing with a stopwatch. Stop timing when the plane has completed 5 rotations around its axis of rotation, recording the time it takes to traverse that distance in the data table. Repeat flying plane in circle for 5 trials.

Data

Mass of Plane (kg) (±0.0001kg)	Length of String (m) (±0.001m)	Trial 1 (s) (±0.01s)	Trial 2 (s) (±0.01s)	Trial 3 (s) (±0.01s)	Trial 4 (s) (±0.01s)	Trial 5(s) (±0.01s)
0.2389 kg	1.063 m	7.52 s	7.81 s	7.70 s	7.92 s	7.68 s

Analysis

$t_{avg,\:5\:rotations}=\frac{7.52+7.81+7.70+7.92+7.68}{5}=7.726\:\pm\:0.005\:s$

$t_{avg,\:1\:rotation}=\frac{7.726\:s}{5}=1.545\:\pm0.001\:s$

$F_{net}\:=\:F_t\cos\left(\theta\right)\:\:where\:\theta\:=\:angle\:of\:string\:relative\:to\:horizontal$

$\frac{mv^2}{r}=F_T\cos\left(\theta\right)$

$v=\frac{d}{t}$

$d=2\pi r$

$\frac{m\left(\frac{2\pi r}{t}\right)^2}{rcos\left(\theta\right)}=F_T$

$r=lcos\left(\theta\right)\:where\:l\:=length\:of\:string\:\left(m\right)$

$F_T=\frac{4\pi^2r^2\cdot m}{r\cdot t^2\cdot cos\left(\theta\right)}=\frac{4\pi^2r\cdot m}{t^2\cdot\cos\left(\theta\right)}=\frac{4\pi^2m\cdot l\cdot cos\left(\theta\right)}{t^2\cdot\cos\left(\theta\right)}=\frac{4\pi^2m\cdot l}{t^2}$

$F_T=\frac{4\pi^2\left(0.2389\:\pm0.0001\:kg\right)\left(1.063\:\pm0.001\:m\right)}{\left(1.545\:+0.001\:s\right)^2}=4.200\pm\:0.011\:N$

Recommendation and Justification for Tensile Force

Since 4.2 N is such a low rating for string, and presumably the toy will be used by children, I would recommend a minimum of five times the calculated minimum tensional strength, or 21 N, as the strength of the string. This is because, children will most likely yank on the plane, increasing the tension of the string significantly, and in order to prevent injuries, a designer should increase the minimum rating by a safety factor. In this case, a reasonable safety factor would be 5 because it wouldn't add much cost to the string in manufacturing but could reduce the risk of injury immensely.

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[{:section_type=>"rich_text", :content=>"Assumptions\n<ul>\n<li>Plane flies in a circular path</li>\n<li>Plane flies at the same height around its height</li>\n<li>Tension on string remains constant</li>\n<li>Velocity of plane is constant</li>\n<li>Length of string does not change while rotating</li>\n</ul>Procedure\nFirst, take off the plane from the string to which it's attached. Measure the mass of the plane using a balance and record in data table. Then, reattach the plane to the string. Now, measure the length of the string, from the point at which the string is attached to the plane, to the point on the ceiling to which the string is attached. Record the length in data table. Turn on the plane's propeller. Start to fly the plane in a circle. Begin timing with a stopwatch. Stop timing when the plane has completed 5 rotations around its axis of rotation, recording the time it takes to traverse that distance in the data table. Repeat flying plane in circle for 5 trials. \nData\n<table style=\"height: 50px; width: 971px;\" border=\"1\"><tbody>\n<tr>\n<td>Mass of Plane (kg) (±0.0001kg)\n</td>\n<td>Length of String (m) (±0.001m)</td>\n<td>Trial 1 (s) (±0.01s)</td>\n<td>Trial 2 (s) (±0.01s)</td>\n<td>Trial 3 (s) (±0.01s)</td>\n<td>Trial 4 (s) (±0.01s)</td>\n<td>Trial 5(s) (±0.01s)</td>\n</tr>\n<tr>\n<td>0.2389 kg</td>\n<td>1.063 m</td>\n<td>7.52 s</td>\n<td>7.81 s</td>\n<td>7.70 s</td>\n<td>7.92 s</td>\n<td>7.68 s</td>\n</tr>\n</tbody></table> \nAnalysis\n<img src=\"/files/30153143/download?download_frd=1&amp;verifier=2hyDySdZd7MhDw9hB20JmjoeXIkicl5stCmKMFpX\" alt=\"FBD.png\" width=\"209\" height=\"226\">\n<img class=\"equation_image\" title=\"t_{avg,\\:5\\:rotations}=\\frac{7.52+7.81+7.70+7.92+7.68}{5}=7.726\\:\\pm\\:0.005\\:s\" src=\"/equation_images/t_%257Bavg%252C%255C%253A5%255C%253Arotations%257D%253D%255Cfrac%257B7.52%2B7.81%2B7.70%2B7.92%2B7.68%257D%257B5%257D%253D7.726%255C%253A%255Cpm%255C%253A0.005%255C%253As\" alt=\"t_{avg,\\:5\\:rotations}=\\frac{7.52+7.81+7.70+7.92+7.68}{5}=7.726\\:\\pm\\:0.005\\:s\">\n<img class=\"equation_image\" title=\"t_{avg,\\:1\\:rotation}=\\frac{7.726\\:s}{5}=1.545\\:\\pm0.001\\:s\" src=\"/equation_images/t_%257Bavg%252C%255C%253A1%255C%253Arotation%257D%253D%255Cfrac%257B7.726%255C%253As%257D%257B5%257D%253D1.545%255C%253A%255Cpm0.001%255C%253As\" alt=\"t_{avg,\\:1\\:rotation}=\\frac{7.726\\:s}{5}=1.545\\:\\pm0.001\\:s\">\n<img class=\"equation_image\" title=\"F_{net}\\:=\\:F_t\\cos\\left(\\theta\\right)\\:\\:where\\:\\theta\\:=\\:angle\\:of\\:string\\:relative\\:to\\:horizontal\" src=\"/equation_images/F_%257Bnet%257D%255C%253A%253D%255C%253AF_t%255Ccos%255Cleft%2528%255Ctheta%255Cright%2529%255C%253A%255C%253Awhere%255C%253A%255Ctheta%255C%253A%253D%255C%253Aangle%255C%253Aof%255C%253Astring%255C%253Arelative%255C%253Ato%255C%253Ahorizontal\" alt=\"F_{net}\\:=\\:F_t\\cos\\left(\\theta\\right)\\:\\:where\\:\\theta\\:=\\:angle\\:of\\:string\\:relative\\:to\\:horizontal\">\n<img class=\"equation_image\" title=\"\\frac{mv^2}{r}=F_T\\cos\\left(\\theta\\right)\" src=\"/equation_images/%255Cfrac%257Bmv%255E2%257D%257Br%257D%253DF_T%255Ccos%255Cleft%2528%255Ctheta%255Cright%2529\" alt=\"\\frac{mv^2}{r}=F_T\\cos\\left(\\theta\\right)\">\n<img class=\"equation_image\" title=\"v=\\frac{d}{t}\" src=\"/equation_images/v%253D%255Cfrac%257Bd%257D%257Bt%257D\" alt=\"v=\\frac{d}{t}\">\n<img class=\"equation_image\" title=\"d=2\\pi r\" src=\"/equation_images/d%253D2%255Cpi%2520r\" alt=\"d=2\\pi r\">\n<img class=\"equation_image\" title=\"\\frac{m\\left(\\frac{2\\pi r}{t}\\right)^2}{rcos\\left(\\theta\\right)}=F_T\" src=\"/equation_images/%255Cfrac%257Bm%255Cleft%2528%255Cfrac%257B2%255Cpi%2520r%257D%257Bt%257D%255Cright%2529%255E2%257D%257Brcos%255Cleft%2528%255Ctheta%255Cright%2529%257D%253DF_T\" alt=\"\\frac{m\\left(\\frac{2\\pi r}{t}\\right)^2}{rcos\\left(\\theta\\right)}=F_T\">\n<img class=\"equation_image\" title=\"r=lcos\\left(\\theta\\right)\\:where\\:l\\:=length\\:of\\:string\\:\\left(m\\right)\" src=\"/equation_images/r%253Dlcos%255Cleft%2528%255Ctheta%255Cright%2529%255C%253Awhere%255C%253Al%255C%253A%253Dlength%255C%253Aof%255C%253Astring%255C%253A%255Cleft%2528m%255Cright%2529\" alt=\"r=lcos\\left(\\theta\\right)\\:where\\:l\\:=length\\:of\\:string\\:\\left(m\\right)\">\n<img class=\"equation_image\" title=\"F_T=\\frac{4\\pi^2r^2\\cdot m}{r\\cdot t^2\\cdot cos\\left(\\theta\\right)}=\\frac{4\\pi^2r\\cdot m}{t^2\\cdot\\cos\\left(\\theta\\right)}=\\frac{4\\pi^2m\\cdot l\\cdot cos\\left(\\theta\\right)}{t^2\\cdot\\cos\\left(\\theta\\right)}=\\frac{4\\pi^2m\\cdot l}{t^2}\" src=\"/equation_images/F_T%253D%255Cfrac%257B4%255Cpi%255E2r%255E2%255Ccdot%2520m%257D%257Br%255Ccdot%2520t%255E2%255Ccdot%2520cos%255Cleft%2528%255Ctheta%255Cright%2529%257D%253D%255Cfrac%257B4%255Cpi%255E2r%255Ccdot%2520m%257D%257Bt%255E2%255Ccdot%255Ccos%255Cleft%2528%255Ctheta%255Cright%2529%257D%253D%255Cfrac%257B4%255Cpi%255E2m%255Ccdot%2520l%255Ccdot%2520cos%255Cleft%2528%255Ctheta%255Cright%2529%257D%257Bt%255E2%255Ccdot%255Ccos%255Cleft%2528%255Ctheta%255Cright%2529%257D%253D%255Cfrac%257B4%255Cpi%255E2m%255Ccdot%2520l%257D%257Bt%255E2%257D\" alt=\"F_T=\\frac{4\\pi^2r^2\\cdot m}{r\\cdot t^2\\cdot cos\\left(\\theta\\right)}=\\frac{4\\pi^2r\\cdot m}{t^2\\cdot\\cos\\left(\\theta\\right)}=\\frac{4\\pi^2m\\cdot l\\cdot cos\\left(\\theta\\right)}{t^2\\cdot\\cos\\left(\\theta\\right)}=\\frac{4\\pi^2m\\cdot l}{t^2}\">\n<img class=\"equation_image\" title=\"F_T=\\frac{4\\pi^2\\left(0.2389\\:\\pm0.0001\\:kg\\right)\\left(1.063\\:\\pm0.001\\:m\\right)}{\\left(1.545\\:+0.001\\:s\\right)^2}=4.200\\pm\\:0.011\\:N\" src=\"/equation_images/F_T%253D%255Cfrac%257B4%255Cpi%255E2%255Cleft%25280.2389%255C%253A%255Cpm0.0001%255C%253Akg%255Cright%2529%255Cleft%25281.063%255C%253A%255Cpm0.001%255C%253Am%255Cright%2529%257D%257B%255Cleft%25281.545%255C%253A%2B0.001%255C%253As%255Cright%2529%255E2%257D%253D4.200%255Cpm%255C%253A0.011%255C%253AN\" alt=\"F_T=\\frac{4\\pi^2\\left(0.2389\\:\\pm0.0001\\:kg\\right)\\left(1.063\\:\\pm0.001\\:m\\right)}{\\left(1.545\\:+0.001\\:s\\right)^2}=4.200\\pm\\:0.011\\:N\">\nRecommendation and Justification for Tensile Force\nSince 4.2 N is such a low rating for string, and presumably the toy will be used by children, I would recommend a minimum of five times the calculated minimum tensional strength, or 21 N, as the strength of the string. This is because, children will most likely yank on the plane, increasing the tension of the string significantly, and in order to prevent injuries, a designer should increase the minimum rating by a safety factor. In this case, a reasonable safety factor would be 5 because it wouldn't add much cost to the string in manufacturing but could reduce the risk of injury immensely."}]

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Physics Toys, Inc. Tension Analysis

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