Power- 30th jan'16

Question:

A force acting on a body depends on its displacement !!S!! as
!!F prop S^(-2/3)!!. The power delivered by !!F!! will depend on displacement as:

(a) !!S^(2/3)!!

(b) !!S^(5/6)!!

(c) !!S^(1/6)!!

(d) None

Solution

!!F prop S^(-2/3) => F= K S^(-2/3)....(1)!!

!!F= ma!!

!!=> ma = K S^(-2/3)!!

!!=> a = K/m S^(-2/3)!! and !!a = v (dv)/(ds)!!

!!=> v (dv)/(ds) = K S^(-2/3)!!

!!=> v dv = K S^(-2/3) ds!!

Integrating both teh sides,

!!=> int_0^v v dv = K/m int_0^S S^(-2/3) ds!!

!!=> v^2/2 = (K/m) (3S^(1/3))!!

!!=> v^2 = (6K/m) S^(1/3)!!

!!=> v = sqrt(6K/m) S^(1/6).....(2)!!

Power !!,P= F v!!

From equation !!(1)!! and !!(2)!!, we get

!!P= K S^(-2/3) (sqrt(6K/m) S^(1/6))!!

!!=> P = K sqrt(6K/m) S^(-1/2)!!

!!P prop S^(-1/2)!!

Therefore, option (d) is correct.