Find the total work done in moving a particle in a force field given by F=3xyi-5zj+10xk along the curve x=t^2+1, y=2t^2, z=t^3 from t=1 to t=2.
Given,
"F=(3xyi-5zj+10xk)\\\\x=t^2+1\\\\y=2t^2\\\\z=t^3"
We have,
"\\bar F\\cdot \\bar{dr}=(3xyi-5zj+10xk)\\cdot(dxi+dyj+dzk)"
"= 3xydx-5zdy+10xdz\\\\=3(t^2+1)\\cdot(2t^2)(2tdt)-5(t^3)(4tdt)+10(t^2+1)(3t^2dt)\\\\=(12t^5+12t^3-20t^4+30t^4+30t^2)dt\\\\=(12t^5+10t^4+12t^3+30t^2)dt"
So, Work Done = "\\int_c \\bar F\\cdot \\bar{dr}" "=\\int_1^2(12t^5+10t^4+12t^3+30t^2)dt\\\\"
"\\\\=[\\frac{12t^6}{5}+\\frac{10t^5}{5}+\\frac{12t^4}{4}+\\frac{30t^3}{3}]_1^2\\\\=[2t^6+2t^5+3t^4+10t^3]_1^2\\\\=[2(2)^6+2(2)^5+3(2)^4+10(2)^3]-[2+2+3+10]\\\\=303"
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