SATHEE: Properties of Matter 2 Question 19

Properties of Matter 2 Question 19

19. A uniform solid cylinder of density $0.8 g / c m^{3}$ floats in equilibrium in a combination of two non-mixing liquids $A$ and $B$ with its axis vertical. The densities of the liquids $A$ and $B$ are $0.7 g / c m^{3}$ and $1.2 g / c m^{3}$ , respectively. The height of liquid $A$ is $h_{A} = 1.2 c m$ . The length of the part of the cylinder immersed in liquid $B$ is $h_{B} = 0.8 c m$ .

(2002, 5M)

(a) Find the total force exerted by liquid $A$ on the cylinder.

(b) Find $h$ , the length of the part of the cylinder in air.

(c) The cylinder is depressed in such a way that its top surface is just below the upper surface of liquid $A$ and is then released. Find the acceleration of the cylinder immediately after it is released.

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Answer:

Correct Answer: 19. (a) Zero (b) 0.25 cm (c) g / 6

Solution:

(a) Liquid $A$ is applying the hydrostatic force on cylinder from all the sides. So, net force is zero.

(b) In equilibrium

Weight of cylinder $=$ Net upthrust on the cylinder

Let $s$ be the area of cross-section of the cylinder, then weight $= (s) (h + h_{A} + h_{B}) ρ_{cylinder} g$

and upthrust on the cylinder

= upthrust due to liquid $A +$ upthrust due to liquid $B$

$= s h_{A} ρ_{A} g + s h_{B} ρ_{B} g$

Equating these two,

$s (h + h_{A} + h_{B}) ρ_{cylinder} g = s g (h_{A} ρ_{A} + h_{B} ρ_{B})$

or $(h + h_{A} + h_{B}) ρ_{cylinder} = h_{A} ρ_{A} + h_{B} ρ_{B}$

Substituting,

$h_{A} = 1.2 c m, h_{B} = 0.8 c m$ and $ρ_{A} = 0.7 g / c m^{3}$

$ρ_{B} = 1.2 g / c m^{3}$ and $ρ_{cylinder} = 0.8 g / c m^{3}$

In the above equation, we get $h = 0.25 c m$

$∴$ Net acceleration $a = \frac{Force}{Mass of cylinder}$

$\begin{aligned} or & a = \frac{s h ρ_{B} g}{s (h + h_{A} + h_{B}) ρ_{cylinder}} \\ or & a = \frac{h ρ_{B} g}{(h + h_{A} + h_{B}) ρ_{cylinder}} \end{aligned}$

Substituting the values of $h, h_{A} h_{B}, ρ_{B}$ and $ρ_{cylinder}$ , we get, $a = \frac{g}{6} (u p w a r d s)$

Properties of Matter 2 Question 19

Answer:

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Properties of Matter 2 Question 19

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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