Find the work done by the force field
Force > Work Done
Question
Find the work done by the force field →F({x,y,z})= -12xˆi-12yˆj+14ˆk on a particle as it moves along the helix →r(t)=costˆi+sintˆj+tˆk from point (1,0,0) to point (( -1,0,3π ).
Solution:
Here
→F(x,y,z)= -12xˆi-12yˆj+14ˆk - - -- - -(A)
And
→r(t)=costˆi+sintˆj+tˆk
We know
that
→r(t)=xˆi+yˆj+zˆk
So,
x=cost , y = sint , z = t
By taking derivative w.r.t t of →r(t)
→dr(t)=(-sintˆi+cosˆj+ˆk)dt- --- -(1)
By taking
dot product of (A) and (1)
→F.d→r(t)=(-12costˆi-12sintˆj + 14ˆk).(-sintˆI +costˆj + ˆk)dt
→F.d→r(t)=(12cost.sint-12sint.cost+14)dt
→F.d→r(t)=14dt
W=small∫→F.d→r(t)
Total work done = W = 14[∫-11dt+∫3π0dt]
= 14[(1+1)+(0-3Ï€ )]
= 14[2-3Ï€ ]
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