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Catégorie :Category: nCreator TI-Nspire
Auteur Author: oONOLTZOo
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 2.70 Ko KB
Mis en ligne Uploaded: 10/10/2024 - 12:45:13
Uploadeur Uploader: oONOLTZOo (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4245397
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 2.70 Ko KB
Mis en ligne Uploaded: 10/10/2024 - 12:45:13
Uploadeur Uploader: oONOLTZOo (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4245397
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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Luigi Tavolino, furniture designer, conceives of alightweight table of daring simplicity: a flat sheet of toughened glass supported on slender, unbraced cylindrical legs(Figure 1). The legs must be solid (to make them thin) and as light as possible (to makethe table easier to move). They must support the tabletop and whatever is placed upon itwithout buckling. What materials could one recommend? Let's break down the steps you've provided in yourrearrangement for the material index of solid cylindrical legs, clarifying thereasoning behind each transformation. 1. Starting with the Equation You began with the equation for the force acting on a solidcylindrical leg: F= n^2 * À^2 *E * I / L2 Given n=1n, the equation simplifies to: F=À^2 * E* I / L^2 2. Moment of Inertia I The moment of inertia III for a solid cylinder is given by: I= (À / 4) *r^4 Substituting this into the equation for FFF: F= À^2 E (À/4 *r^4) / L^2 This simplifies to: F= À^3 * Er^4 / 4*L^2 3. Isolating r^4 To isolate r4r^4r4, multiply both sides by 4L^2 4 * L^2 * F = À^3 * E* r^4 Now, divide both sides by À^3E r4=4L^2 * F / À^3*E 4. Mass Equation The mass m of the cylindrical leg is given by the volumetimes density: m=A * L * density For a solid cylinder, the cross-sectional area A is: A=À*r^2 So we can rewrite the mass equation as: m=À*r2*L Å density 5. Isolating r2r^2r2 To isolate r^2 rearrange the equation: R^2= m / À * L * density This can be rewritten as: R^2=m/À * 1 / L * density 6. Expressing rrr Taking the square root of both sides gives us: r= sqrt(m À) * 1 / sqrt (L * density ) 7. Substituting r4r^4r4 into Mass Equation From r^4=4* L^2 * F / À^3*E, we can express m: Substituting r^2 back into the equation for r^4: R^4=(m / À* L * density ) ^ 2 This leads to: r^4=m^2 / À^2 * L^2 * density^2 8. Equating the Two Expressions for r^4 Set the two expressions for r^4 equal: M^2 / À^2 *L^2 * density^2 = 4 * L^2 * F / À^ 3 * E 9. Rearranging for m^2 Multiply both sides by À^2 * L^2 * density^2 M^2= 4* L^4 * F * density2 / À^ 5 * E 10. Taking the Square Root Taking the square root of both sides gives us: M^2= sqrt(4*L^4 * F * density^2 / À^ 5 * E) This simplifies to: m=2*L^2 * density * sqrt( F) /sqrt( À^ 5 * E) 11. Final Rearrangement If you take out the constants, you can express mmm in termsof E and density M = 4L^4 * density^2 * F/ À^ 5 * E This leads to your conclusion: M= E^1/2 / density Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
Luigi Tavolino, furniture designer, conceives of alightweight table of daring simplicity: a flat sheet of toughened glass supported on slender, unbraced cylindrical legs(Figure 1). The legs must be solid (to make them thin) and as light as possible (to makethe table easier to move). They must support the tabletop and whatever is placed upon itwithout buckling. What materials could one recommend? Let's break down the steps you've provided in yourrearrangement for the material index of solid cylindrical legs, clarifying thereasoning behind each transformation. 1. Starting with the Equation You began with the equation for the force acting on a solidcylindrical leg: F= n^2 * À^2 *E * I / L2 Given n=1n, the equation simplifies to: F=À^2 * E* I / L^2 2. Moment of Inertia I The moment of inertia III for a solid cylinder is given by: I= (À / 4) *r^4 Substituting this into the equation for FFF: F= À^2 E (À/4 *r^4) / L^2 This simplifies to: F= À^3 * Er^4 / 4*L^2 3. Isolating r^4 To isolate r4r^4r4, multiply both sides by 4L^2 4 * L^2 * F = À^3 * E* r^4 Now, divide both sides by À^3E r4=4L^2 * F / À^3*E 4. Mass Equation The mass m of the cylindrical leg is given by the volumetimes density: m=A * L * density For a solid cylinder, the cross-sectional area A is: A=À*r^2 So we can rewrite the mass equation as: m=À*r2*L Å density 5. Isolating r2r^2r2 To isolate r^2 rearrange the equation: R^2= m / À * L * density This can be rewritten as: R^2=m/À * 1 / L * density 6. Expressing rrr Taking the square root of both sides gives us: r= sqrt(m À) * 1 / sqrt (L * density ) 7. Substituting r4r^4r4 into Mass Equation From r^4=4* L^2 * F / À^3*E, we can express m: Substituting r^2 back into the equation for r^4: R^4=(m / À* L * density ) ^ 2 This leads to: r^4=m^2 / À^2 * L^2 * density^2 8. Equating the Two Expressions for r^4 Set the two expressions for r^4 equal: M^2 / À^2 *L^2 * density^2 = 4 * L^2 * F / À^ 3 * E 9. Rearranging for m^2 Multiply both sides by À^2 * L^2 * density^2 M^2= 4* L^4 * F * density2 / À^ 5 * E 10. Taking the Square Root Taking the square root of both sides gives us: M^2= sqrt(4*L^4 * F * density^2 / À^ 5 * E) This simplifies to: m=2*L^2 * density * sqrt( F) /sqrt( À^ 5 * E) 11. Final Rearrangement If you take out the constants, you can express mmm in termsof E and density M = 4L^4 * density^2 * F/ À^ 5 * E This leads to your conclusion: M= E^1/2 / density Made with nCreator - tiplanet.org
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