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Find the work done by the force field f(x,y,z)=3xi+3yj+7kf(x,y,z)=3xi+3yj+7k

Find the work done by the force field f(x,y,z)=3xi+3yj+7kf(x,y,z)=3xi+3yj+7k on a particle that moves along the helix r(t)=3cos(t)i+3sin(t)j+3tk,0≤t≤2π

2 months ago

Solution 1

Guest Guest #263
2 months ago
Call the path \mathcal C. Then the work done by \mathbf f(x,y,z) along \mathcal C is given by the line integral,\displaystyle\int_{\mathcal C}\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=2\pi}\mathbf f(x(t),y(t),z(t))\cdot\dfrac{\mathrm d\mathbf r}{\mathrm dt}\,\mathrm dt
Swapping the \mathbf{ijk} notation out for ordered component notation, I'll write
\mathbf r(t)=(x(t),y(t),z(t))=(3\cos t,3\sin t,3t)
so that
\dfrac{\mathrm d\mathbf r}{\mathrm dt}=(-3\sin t,3\cos t,3)
The line integral reduces to\displaystyle\int_0^1(9\cos t,9\sin t,7)\cdot(-3\sin t,3\cos t,3)\,\mathrm dt=\displaystyle\int_0^1(-27\cos t\sin t+27\sin t\cos t+21)\,\mathrm dt=\displaystyle21\int_0^1\mathrm dt=21

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