Drag on the Sphere

Classical Mechanics Level 4

A sphere of radius r r is abandoned in a chamber of height H H ( H H much greater than r r ) filled with a fluid with viscosity η \eta and density ρ \rho . According to the Stoke's Law, for small velocities and laminar flow, the frictional force exerted by the viscous fluid in the sphere is given by the expression F d ⃗ = − 6 π r η v ⃗ \vec { { F }_{ d } } =-6\pi r\eta \vec { v } . If W W is the total work done by frictional force and buyoancy until the sphere reaches the height h h in which the resultant force is zero, the correct expression for h h is...

Details and Assumptions :

  • Use m m for sphere's mass and g g for the gravitational acceleration.

  • Assume also that the time elapsed is infinitely long, so that the sphere's velocity actually reaches the terminal velocity.

H + W m g H+\frac { W }{ mg } H + W / ( m g ) − g 2 ( 4 π ρ r 3 − 3 m 18 η π r ) 2 H+W/(mg)-\frac{g}{2}\left( \frac { 4\pi \rho { r }^{ 3 }-3m }{ 18\eta \pi r } \right) ^{ 2 } H + W m g − 1 2 g ( 3 m g 18 η π r ) 2 H+\frac{W}{ mg } -\frac{1}{2g}\left( \frac {3mg }{ 18\eta \pi r } \right) ^{ 2 } H − W m g − 1 2 g ( 3 m g 18 η π r ) 2 H-\frac{W}{ mg } -\frac{1}{2g}\left( \frac {3mg }{ 18\eta \pi r } \right) ^{ 2 } H − W m g H-\frac { W }{ mg } H − W / ( m g ) − g 2 ( 4 π ρ r 3 − 3 m 18 η π r ) 2 H-W/(mg) -\frac{g}{2}\left( \frac { 4\pi \rho { r }^{ 3 }-3m }{ 18\eta \pi r } \right)^{ 2 }

João Vitor Cordeiro de Brito by João Vitor Cordeiro de B… , 27

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