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Terminal Velocity

notes

Terminal Velocity

  • Terminal velocity is the maximum velocity of a body moving through a viscous fluid.

  • It is attained when the force of resistance of the medium is equal and opposite to the force of gravity.

  • As the velocity is increasing the retarding force will also increase and a stage will come when the force of gravity becomes equal to the resistance force.

  • After that point velocity won’t increase and this velocity is known as terminal velocity.

  • It is denoted by ‘vt’.Wheret=terminal.

  • Mathematically:-

  • Terminal velocity is attained when the Force of resistance = force due to gravitational attraction.

`6pieta"rv"="mg"`

`6pieta"rv"="density"xx"V"_ g "(Because density"="m"/"V")`, density = `rho-sigma` where `rho` and `sigma` are the densities of the sphere and the viscous medium.

`6pieta"rv" = (rho-sigma)xx4/3pir^3g` where volume of the sphere(V)=`4/3pir^3`

By simplifying

`=(rho-sigma)"g"xx4/3r^2xx1/(6eta)`

`v_t=(2r^2(rho-sigma)g)/(9eta)`This is the terminal velocity. Where`(v=v_t)`

Problem: The terminal velocity of a copper ball of radius 2.0 mm falling through a tank of oil at 20oC is 6.5 cm s-1.Compute the viscosity of the oil at 20oC.Density of oil is 1.5 × 103 kg m-3, density of copper is 8.9 × 103 kg m-3.

Solution:

Given:`"V"_t=6.5xx10^-2ms^-1, a=2xx10^-3m,g=9.8  ms^-2, rho=8.9xx10^3kgm^-3, sigma=1.5xx10^3kgm^-3.`

From equation:` -v_t=(2r^2(rho-sigma)g)/(9eta)`

`=2/9((2xx10^-3m^2xx9.8ms^-2)/(6.5xx10^-2ms^-1))xx7.4xx10^3"kgm"^-3`

`=9.9xx10^-1"kgm"^-1"s"^-1`

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