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Properties of Matter question

2006 · Shift 0 · Q105
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Properties of Matter question

2006 · Shift 0 · Q105

JEE MainPhysicsProperties of MatterMCQ+4 / −1
If the terminal speed of a sphere of gold (density =19.5  kg/m3= 19.5\,\,kg/{m^3}=19.5kg/m3) is 0.2m/s0.2m/s0.2m/s in a viscous liquid (density =1.5  kg/m3= 1.5\,\,kg/{m^3}=1.5kg/m3, find the terminal speed of a sphere of silver (density =10.5  kg/m3= 10.5\,\,kg/{m^3}=10.5kg/m3) of the same size in the same liquid
  1. A
    0.4m/s0.4m/s0.4m/s
  2. B
    0.133m/s0.133m/s0.133m/s
  3. C
    0.1m/s0.1m/s0.1m/s
  4. D
    0.2m/s0.2m/s0.2m/s
View written solutionFree

Correct answer: C

  1. Use Stokes' law for terminal speed

For a sphere of radius rrr falling in a viscous liquid, terminal speed is

vt=2r2g(ρs−ρl)9ηv_t = \frac{2r^2 g (\rho_s - \rho_l)}{9\eta}vt​=9η2r2g(ρs​−ρl​)​

where:

  • ρs\rho_sρs​ = density of sphere
  • ρl\rho_lρl​ = density of liquid
  • η\etaη = viscosity of liquid
  1. Identify proportionality

Since both spheres have:

  • same size ⇒r\Rightarrow r⇒r same
  • same liquid ⇒ρl\Rightarrow \rho_l⇒ρl​ same
  • same viscosity ⇒η\Rightarrow \eta⇒η same

So,

vt∝(ρs−ρl)v_t \propto (\rho_s - \rho_l)vt​∝(ρs​−ρl​)

Hence,

vsilvervgold=ρsilver−ρlρgold−ρl\frac{v_{\text{silver}}}{v_{\text{gold}}} = \frac{\rho_{\text{silver}} - \rho_l}{\rho_{\text{gold}} - \rho_l}vgold​vsilver​​=ρgold​−ρl​ρsilver​−ρl​​

  1. Substitute values

Given:

vgold=0.2 m/sv_{\text{gold}} = 0.2\,\text{m/s}vgold​=0.2m/s ρgold=19.5\rho_{\text{gold}} = 19.5ρgold​=19.5 ρsilver=10.5\rho_{\text{silver}} = 10.5ρsilver​=10.5 ρl=1.5\rho_l = 1.5ρl​=1.5

So,

vsilver=0.2×10.5−1.519.5−1.5v_{\text{silver}} = 0.2 \times \frac{10.5 - 1.5}{19.5 - 1.5}vsilver​=0.2×19.5−1.510.5−1.5​

vsilver=0.2×918v_{\text{silver}} = 0.2 \times \frac{9}{18}vsilver​=0.2×189​

vsilver=0.2×0.5=0.1 m/sv_{\text{silver}} = 0.2 \times 0.5 = 0.1\,\text{m/s}vsilver​=0.2×0.5=0.1m/s

  1. Check options
  • A: 0.4 m/s0.4\,\text{m/s}0.4m/s ❌
  • B: 0.133 m/s0.133\,\text{m/s}0.133m/s ❌
  • C: 0.1 m/s0.1\,\text{m/s}0.1m/s ✅
  • D: 0.2 m/s0.2\,\text{m/s}0.2m/s ❌

Therefore, the correct option is C.

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