A cylindrical wooden float whose base area is S and the height H floats on wide water surface. Density of wood d and density of water is $ρ$ . What minimum work must be performed to take the float out of the water?

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

$\frac{S^{2} g d^{2} H^{2}}{ρ}$

$\frac{S^{2} g d}{2 ρ}$

$\frac{S g d^{2} H^{2}}{2 ρ}$

$\frac{2 S^{3} d^{2} H^{2}}{ρ H}$

answer is C.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

If h height of float is submerged in water, we use

$m g = F_{B} d S H \times g = ρ_{w} S h \times g h = \frac{d H}{ρ}$

Now when force F is applied, for minimum work we slowly pull it out and when submerged length is x, we use

$\begin{array}{l} F - m g + ρ S x g = 0 \\ F = m g - ρ S x g \\ W = \int F d x = \int (m g - ρ S x g) d x \\ = m g \int d x - ρ S g \int x d x \\ W = m g h - \frac{ρ S g h^{2}}{2} \\ = (ρ S h) g h - \frac{ρ S g h^{2}}{2} = \frac{ρ S g h^{2}}{2} \\ \frac{ρ S g}{2} (\frac{d H}{ρ}) = \frac{S g d^{2} H^{2}}{2} \end{array}$