Find the work done by the force field

Author - Learn With Samrah

November 22, 20222 minute read

Force > Work Done

Question No. 1

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 = $\frac{1}{4}[\int_{1}^{-1}dt + \int_{0}^{3pi}dt]$

= $\frac{1}{4}\left[ \left( 1 + 1 \right) + \left( 0 - 3\pi \right) \right]$

= $\frac{1}{4}\left[ 2 - 3\pi \right]$