Question

a force F acting on the body depends on its displacement S as F inversely S⅓.the power delivered by F will depend on displacement a s

Answer 1

power is inversely proportional to S1/2

Answer 2

P is inversely proportional to $S^{(1 / 2)}$.

Given:

Force = F

Displacement = S

$F \propto \frac{1}{\sqrt[3]{S}}$

Solution:  

Force can be written as:

$F=\frac{k}{\sqrt[3]{S}}=\frac{k}{S^{(1 / 3)}}$

In the drag, F is directly proportional to velocity with a power square.

$F \propto v^{2}$

Thereby, the velocity will be proportional to square root of F.

$v \propto \sqrt{F}$

From initial F equation, we will get,  

$v=l \sqrt{F}=l F^{(1 / 2)}$

Where, l is the proportionality constant like k.

$v=l\left(\frac{1}{S^{(1 / 3)}}\right)^{(1 / 2)}$

$v=l\left(\frac{1}{S^{(1 / 6)}}\right)$

We know the formula of power,

$P=\frac{F}{v}$

On substituting, velocity and force, we get,

$\Rightarrow P=\frac{\frac{k}{S^{(1 / 3)}}}{\left(\frac{l}{S(1 / 6)}\right)}$

On taking out the constants, we get,

$\Rightarrow P \propto \frac{\frac{1}{S^{(1 / 3)}}}{\frac{1}{S^{(1 / 6)}}}$

$P \propto \frac{1}{S^{(1 / 3)+(1 / 6)}}$

$P \propto \frac{1}{S^{\left(\frac{2+1}{6}\right)}}$

$P \propto \frac{1}{S^{(3 / 6)}}$

$\Rightarrow P \propto \frac{1}{S(1 / 2)}$

Thereby,

P is inversely proportional to $S^{(1 / 2)}$.