# 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

Mathematics

## Question

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

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
$$\mathbf r(t)=(x(t),y(t),z(t))=(3\cos t,3\sin t,3t)$$
$$\dfrac{\mathrm d\mathbf r}{\mathrm dt}=(-3\sin t,3\cos t,3)$$
The line integral reduces to$$\displaystyle\int_0^1(9\cos t,9\sin t,7)\cdot(-3\sin t,3\cos t,3)\,\mathrm dt$$$$=\displaystyle\int_0^1(-27\cos t\sin t+27\sin t\cos t+21)\,\mathrm dt$$$$=\displaystyle21\int_0^1\mathrm dt=21$$