Consider two solid spheres P and Q each of density 8 gm cm-3 and diameters 1 cm and 0-5 cm respectively- Sphere P is dropped into a liquid of density 0-8 gm cm-3 and viscosity - 3 poiseulles- Sphere Q is dropped into a liquid of density 1-6 gm cm-3 and viscosity - - 2 poiseuille- The ratio of the terminal velocities of P and Q is

Terminal velocity, v_r=2/9r^2/η(ρ σ)g nbsp;, where ρ is the density of the solid sphere and nbsp;σ is the density of the liquid. ∴ v_P/v_Q=(8 0.8)× ...

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Standard VIII

Mathematics

Dimensional Analysis

Consider two ...

Question

Consider two solid spheres $P$ and $Q$ each of density $8 gm c m^{- 3}$ and diameters $1 cm$ and $0.5 cm$ respectively. Sphere $P$ is dropped into a liquid of density $0.8 gm c m^{- 3}$ and viscosity $η = 3$ poiseulles. Sphere $Q$ is dropped into a liquid of density $1.6 gm c m^{- 3}$ and viscosity $η = 2$ poiseuille. The ratio of the terminal velocities of $P$ and $Q$ is

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Solution

Terminal velocity, $v_{r} = \frac{2}{9} \frac{r^{2}}{η} (ρ - σ) g$ , where $ρ$ is the density of the solid sphere and $σ$ is the density of the liquid.
$∴ \frac{v_{P}}{v_{Q}} = \frac{(8 - 0.8) \times {(\frac{1}{2})}^{2} \times 2}{(8 - 1.6) \times {(\frac{1}{4})}^{2} \times 3} = 3$

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Similar questions

Q. Consider two solid spheres

P

and

Q

each of density

8 gm c m^{- 3}

and diameters

1 cm

and

0.5 cm

respectively. Sphere

P

is dropped into a liquid of density

0.8 gm c m^{- 3}

and viscosity

η = 3

poiseulles. Sphere

Q

is dropped into a liquid of density

1.6 gm c m^{- 3}

and viscosity

η = 2

poiseuille. The ratio of the terminal velocities of

P

and

Q

Q. Consider two solid spheres P and Q each of density

8 g

{c m}^{- 3}

and diameters

1

cm and

0.5

cm respectively. Sphere P is dropped into a liquid of density

0.8 g

{c m}^{- 3}

and viscosity

η

=3 poiseulles. Sphere Q is dropped into a liquid of density

1.6 g / c m^{3}

and viscosity

η

=2 poiseulles. The ratio of the terminal velocities of P and Q is:

Q. Two spheres P and Q of equal radii have densities

ρ_{1}

and

ρ_{2}

repectively. The spheres are connected by a massless string and placed in liquids

L_{1}

and

L_{2}

of densities

σ_{1}

and

σ_{2}

and viscosities

η_{1}

and

η_{2}

respectively. They float in equilibrium with the sphere P in

L_{1}

and Q in

L_{2}

and the string being taut (see figure). If sphere P alone in

L_{2}

has terminal velocity

{\to V}_{P}

and Q alone in

L_{1}

has terminal velocity

{\to V}_{Q}

,
(Terminal velocity of a sphere having radius

r

and density

ρ

thrown in a liquid having density

σ

and viscoscity

η

is given by

= \frac{2 r^{2}}{9 η} (ρ - σ)

). Then which of the following are correct ?

Q. Two spheres

P

and

Q

of equal radii have densities

ρ_{1}

and

ρ_{2}

, respectively. The spheres are connected by a massless string and placed in liquids

L_{1}

and

L_{2}

of densities

σ_{1}

and

σ_{2}

and viscosities

η_{1}

and

η_{2}

, respectively. They float in equilibrium with the sphere

P

L_{1}

and sphere

Q

L_{2}

and the string being taut (see figure). If sphere

P

alone in

L_{2}

has terminal velocity

{\to v}_{P}

and

Q

alone in

L_{1}

has terminal velocity

{\to v}_{Q}

, then

Q. Two spheres P and Q of equal radii have densities

ρ_{1}

and

ρ_{2}

, respectively. The spheres are connected by a massless string and placed in liquids

L_{1}

and

L_{2}

of densities

σ_{1}

and

σ_{2}

and viscosities

η_{1}

and

η_{2}

, respectively. They float in equilibrium with the sphere P in

L_{1}

and sphere Q in

L_{2}

and the string being taut. If sphere P along in

L_{2}

has terminal velocity

{\to V}_{P}

and Q along in

L_{1}

has terminal velocity

{\to V}_{Q}

, then

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Terminal velocity, vr=29r2η(ρ−σ)g , where ρ is the density of the solid sphere and σ is the density of the liquid. ∴vPvQ=(8−0.8)×(12)2×2(8−1.6)×(14)2×3=3

Terminal velocity, $v_{r} = \frac{2}{9} \frac{r^{2}}{η} (ρ - σ) g$ , where $ρ$ is the density of the solid sphere and $σ$ is the density of the liquid.
$∴ \frac{v_{P}}{v_{Q}} = \frac{(8 - 0.8) \times {(\frac{1}{2})}^{2} \times 2}{(8 - 1.6) \times {(\frac{1}{4})}^{2} \times 3} = 3$