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Parcial
1.
Centroide es el
punto C, lo cual coincide con el centro de gravedad , punto donde actúa la fuerza resultante.
2.
3.
4.
5.
6.
Primeros momentos Q:
7.
…….Por simetría …….Relación Del grafico
se cumple:
8.
9.
10.
11.
12.
13.
14.
15.
16.
𝑑𝑢 2𝜋𝑢 𝑇𝑒𝑜𝑟𝑒𝑚𝑎 𝑑𝑒 𝑒𝑗𝑒𝑠
𝑃𝑎𝑟𝑎𝑙𝑒𝑙𝑜𝑠 𝑜 𝑆𝑡𝑒𝑖𝑛𝑒𝑟:
17.
18.
19.
𝑑𝐼𝑥𝑦 = 𝑑𝐼𝑥
′𝑦′ + 𝑥 𝑒𝑙 𝑦𝑒𝑙 𝑑𝐴 Solución Integramos para la región 𝑑𝑥 𝑑𝐼𝑥 ′𝑦′ = 0 (𝑠𝑖𝑚𝑒𝑡𝑟𝑖𝑎 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑑𝑖𝑓𝑒𝑟𝑒𝑛𝑐𝑖𝑎𝑙) 𝑥 ′ 𝑦′ 𝑦𝑒𝑙 𝑥 𝑒𝑙 𝑥 𝑒𝑙 = 𝑥 𝑦𝑒𝑙 = 𝑦 2 𝑑𝐴 = 𝑦 𝑑𝑥 𝑑𝐼𝑥𝑦 = 𝑥 𝑦 2 𝑦 𝑑𝑥 = 1 2 𝑥𝑦2 𝑑𝑥 𝑦2 = 4𝑎4 − 𝑥2 𝑎2 4𝑎2 𝑑𝐼𝑥𝑦 = 1 2 𝑥( 4𝑎4 − 𝑥2 𝑎2 4𝑎2 )𝑑𝑥 𝐴 𝑑𝐼𝑥𝑦 = 0 2𝑎 1 2 𝑥 4𝑎4 − 𝑥2 𝑎2 4𝑎2 𝑑𝑥 = 1 8𝑎2 0 2𝑎 (4𝑎4 𝑥 − 𝑥3 𝑎2 )𝑑𝑥 1 8𝑎2 2𝑎4 𝑥2 − 1 4 𝑥4 𝑎2 0 2𝑎 = 1 2 𝑎4
20.
𝑑𝑦 𝑥 ′ 𝑦′ 𝑦𝑒𝑙 𝑥 𝑒𝑙 𝑑𝐼𝑥𝑦
= 𝑑𝐼𝑥 ′𝑦′ + 𝑥 𝑒𝑙 𝑦𝑒𝑙 𝑑𝐴 Integramos para la región 𝑦𝑒𝑙 = 𝑦 𝑥 𝑒𝑙 = 𝑥 2 𝑑𝐴 = 𝑥 𝑑𝑦 𝑑𝐼𝑥𝑦 = 𝑥 2 𝑥 𝑑𝑦 = 1 2 𝑦𝑥2 𝑑𝑦 𝑎, 𝑏 𝑝𝑒𝑟𝑡𝑒𝑛𝑒 𝑎 𝑙𝑎 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 𝑥 = 𝑘𝑦2 → 𝑥 = 𝑎 𝑏2 𝑦2 → 𝑥2 = 𝑎2 𝑏4 𝑦4 𝐴 𝑑𝐼𝑥𝑦 = 0 𝑏 1 2 𝑦 𝑎2 𝑏4 𝑦4 𝑑𝑦 = 𝑎2 2𝑏4 0 𝑏 𝑦5 𝑑𝑦 = 𝑎2 𝑏2 12 Solución 𝑑𝐼𝑥 ′𝑦′ = 0 (𝑠𝑖𝑚𝑒𝑡𝑟𝑖𝑎 𝑑𝑒𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑑𝑖𝑓𝑒𝑟𝑒𝑛𝑐𝑖𝑎𝑙)
21.
22.
23.
24.
25.
𝐼𝑚𝑎𝑥 = 𝐶
+ 𝑅 𝐼𝑚𝑖𝑛 = 𝐶 − 𝑅
26.
𝐼𝑚𝑎𝑥 = 𝐶
+ 𝑅 𝐼𝑚𝑖𝑛 = 𝐶 − 𝑅 𝐶 = 𝐼𝑥 + 𝐼𝑦 2 𝑅 = 𝐼𝑥 − 𝐼𝑦 2 2 + (𝐼𝑥𝑦)2 𝐼𝑥 − 𝐼𝑦 2
27.
El circulo de
Mohr queda definido con el radio y el centro:
28.
29.
30.
31.
32.
33.
El vector par
es un vector libre, por lo tanto, resulta lógico situarlo en un lugar conveniente para los cálculos.
34.
35.
36.
37.
38.
39.
40.
41.
42.
Dirección de F: (vector
unitario) Se cumple:
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