SATHEE: Properties of Matter 2 Question 20

Properties of Matter 2 Question 20

20. A wooden stick of length $L$ , radius $R$ and density $ρ$ has a small metal piece of mass $m$ (of negligible volume) attached to its one end. Find the minimum value for the mass $m$ (in terms of given parameters) that would make the stick float vertically in equilibrium in a liquid of density $σ (> ρ)$ .

(1999, 10M)

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Answer:

Correct Answer: 20. $π R^{2} L (\sqrt{ρ σ} - ρ)$

Solution:

Let $M =$ Mass of stick $= π R^{2} ρ L$

(i)

(ii)

(iii)

$l =$ Immersed length of the rod

$G = C M$ of $rod$

$B =$ Centre of buoyant force $(F)$

$C = C M$ of $rod +$ mass $(m)$

$Y_{C M} =$ Distance of $C$ from bottom of the rod

Mass $m$ should be attached to the lower end because otherwise $B$ will be below $G$ and $C$ will be above $G$ and the torque of the couple of two equal and opposite forces $F$ and $(M + m) g$ will be counter clockwise on displacing the rod leftwards. Therefore, the rod cannot be in rotational equilibrium. See the figure (iii).

Now, refer figures (i) and (ii).

For vertical equilibrium $M g + m g = F$ (upthrust)

or $(π R^{2} L) ρ g + m g = (π R^{2} l) σ g$

$∴ l = \frac{π R^{2} L ρ + m}{π R^{2} σ}$

Position of CM ( of rod $+ m$ ) from bottom

$Y_{C M} = \frac{M \cdot \frac{L}{2}}{M + m} = \frac{(π R^{2} L ρ) \frac{L}{2}}{(π R^{2} L ρ) + m}$

Centre of buoyancy $(B)$ is at a height of $\frac{l}{2}$ from the bottom.

We can see from figure (ii) that for rotational equilibrium of the rod, $B$ should either lie above $C$ or at the same level of $B$ .

Therefore, $\frac{l}{2} \geq Y_{C M}$

$\begin{aligned} or & \frac{π R^{2} L ρ + m}{2 π R^{2} σ} & \geq \frac{(π R^{2} L ρ) \frac{L}{2}}{(π R^{2} L ρ) + m} \\ or & m + π R^{2} L ρ & \geq π R^{2} L \sqrt{ρ σ} \\ or & m & \geq π R^{2} L (\sqrt{ρ σ} - ρ) \end{aligned}$

$∴$ Minimum value of $m$ is $π R^{2} L (\sqrt{ρ σ} - ρ)$ .

Properties of Matter 2 Question 20

Answer:

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Properties of Matter 2 Question 20

20. A wooden stick of length L, radius R and density ρ has a small metal piece of mass m (of negligible volume) attached to its one end. Find the minimum value for the mass m (in terms of given parameters) that would make the stick float vertically in equilibrium in a liquid of density σ(>ρ).

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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