Calculus 3

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Mathematics

Empire Beauty School - Pittsburgh

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MATH Moments of Inertia When this activity is concluded you should be able to: 1. Articulate the theoretical construction of the integration of the moment of inertia about an axis using a double summation. 2. Be able to correctly set up the integration to compute the moment of inertia about an axis. 3. Compute the radius of gyration from an axis. The Moment of Inertia of a particle, mass m, about an axis is defined by: 𝐼 = π‘š 𝑑 2 where d is the distance of the particle to the axis. 1. Draw a region, then slice the region into small rectangles. Identify one of the rectangles βˆ— βˆ— as containing an arbitrarily chosen point, (π‘₯𝑖𝑗 , 𝑦𝑖𝑗 ). Using an approach similar to yesterday’s development of the center of mass, we get: π‘˜ 𝐼π‘₯ = 2 βˆ— βˆ— βˆ— lim βˆ‘ βˆ‘(𝑦𝑖𝑗 ) 𝜌(π‘₯𝑖𝑗 , 𝑦𝑖𝑗 ) βˆ†π΄ = ∬ 𝑦 2 𝜌(π‘₯, 𝑦) 𝑑𝐴 π‘˜,π‘™β†’βˆž 𝐷 𝑖=1 𝑗=1 π‘˜ 𝐼𝑦 = 𝑙 𝑙 2 βˆ— βˆ— βˆ— lim βˆ‘ βˆ‘(π‘₯𝑖𝑗 ) 𝜌(π‘₯𝑖𝑗 , 𝑦𝑖𝑗 ) βˆ†π΄ = ∬ π‘₯ 2 𝜌(π‘₯, 𝑦) 𝑑𝐴 π‘˜,π‘™β†’βˆž 𝐷 𝑖=1 𝑗=1 The moment of inertia about the origin is given by: π‘˜ 𝐼𝑂 = 𝑙 2 2 βˆ— βˆ— βˆ— βˆ— lim βˆ‘ βˆ‘ [(π‘₯𝑖𝑗 ) +(𝑦𝑖𝑗 ) ] 𝜌(π‘₯𝑖𝑗 , 𝑦𝑖𝑗 ) βˆ†π΄ = ∬ (π‘₯ 2 +𝑦 2 ) 𝜌(π‘₯, 𝑦) 𝑑𝐴 π‘˜,π‘™β†’βˆž 𝑖=1 𝑗=1 𝐷 2 2 βˆ— βˆ— 2. Explain precisely from where the factor (π‘₯𝑖𝑗 ) +(𝑦𝑖𝑗 ) comes in the 𝐼𝑂 formula. 3. Add the formulae for 𝐼π‘₯ and 𝐼𝑦 together. What is this equal to? (Call this result β€œEq. 1”). 4. Find the moment of inertia about the origin of the disk, D, centered at the origin, with radius, 𝛼 and homogenously distributed density, 𝜌. (You might-should think about what coordinate system works best in this case). 5. Use eq. 1 (found in problem 3, above) and symmetry to find the moments of inertia about the x and y-axes. The Radius of gyration of a lamina about an axis is R such that π‘šπ‘… 2 = 𝐼, where m is the mass of the lamina, I is the moment of inertia about the axis, and R is the distance from the axis. 6. Find the radius of gyration about the y-axis in the previous problem. (The notation we use for this is: 𝑦̿ ).
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