Force > Work Done
Question No. 1
Find the work done by the force field `\vec F( {x,y,z}) = - \frac{1}{2}x\hat i - \frac{1}{2}y\hat j + \frac{1}{4}\hat k` on a particle as it moves along the helix `\vec r\left( t \right) = cost\hat i + sint\hat j + t\hat k` from point (1,0,0) to point `\(\left( - 1,0,3\pi \right).`
Solution:
Here
`\vec F\left( x,y,z \right) = - \frac{1}{2}x\hat i - \frac{1}{2}y\hat j + \frac{1}{4}\hat k - - - - - - \left( A \right)`
And
`\vec r\left( t \right) = cost\hat i + sint\hat j + t\hat k`
We know that
`\vec r\left( t \right) = x\hat i + y\hat j + z\hat k`
So,
x=cost , y = sint , z = t
By taking derivative w.r.t t of `\vec r\left( t \right)`
`\vec{dr} \left( t \right) = \left( - sint\hat i + cos\hat j + \hat k \right)dt - - - - - \left( 1 \right)`
By taking dot product of (A) and (1)
`\vec F .d\vec r\left( t \right) = ( - \frac{1}{2}cost\hat i - \frac{1}{2}sint\hat j + \frac{1}{4}\hat k ) . ( - sint\hat I + cost\hat j + \hat k)dt`
`\vec F . d\vec r\left( t \right) = \left( \frac{1}{2}cost . sint - \frac{1}{2}sint . cost + \frac{1}{4} \right)dt`
`\vec F . d\vec r\left( t \right) = \frac{1}{4}dt`
`W = \smallint \vec F .d\vec r\left( t \right)`
Total work done = W = `\frac{1}{4}[\int_{1}^{-1}dt + \int_{0}^{3pi}dt] `
= `\frac{1}{4}\left[ \left( 1 + 1 \right) + \left( 0 - 3\pi \right) \right]`
= `\frac{1}{4}\left[ 2 - 3\pi \right]`