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Thread: forces again

  1. #1
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    forces again

    When an object moves through a fluid , it experiences a retarding force due to turbulence . For a sphere of radius r moving with a speed of v in a fluid of density , p , the retarding force is given by

    $\displaystyle
    F=kpr^2v^2
    $

    By relating the retarding force to the transfer of momentum between the fluid and the sphere , explain why the force F is directly proportional to $\displaystyle pr^2v^2$
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  2. #2
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    Hello thereddevils
    Quote Originally Posted by thereddevils View Post
    When an object moves through a fluid , it experiences a retarding force due to turbulence . For a sphere of radius r moving with a speed of v in a fluid of density , p , the retarding force is given by

    $\displaystyle
    F=kpr^2v^2
    $

    By relating the retarding force to the transfer of momentum between the fluid and the sphere , explain why the force F is directly proportional to $\displaystyle pr^2v^2$
    In a small time $\displaystyle \delta t$, the sphere moves a distance $\displaystyle \delta x$.We now assume that a volume of fluid proportional to the surface area of the sphere multiplied by the distance $\displaystyle \delta x$ has its velocity changed from zero to the velocity, $\displaystyle v$, of the sphere, due to impact with the sphere's surface. Since the SA of the sphere is proportional to the square of its radius, the mass of fluid affected in time $\displaystyle \delta t$ is therefore equal to

    $\displaystyle k \rho r^2 \delta x$

    where $\displaystyle \rho$ is the density of the fluid, and $\displaystyle k$ is a constant.

    Its change in momentum is therefore

    $\displaystyle k \rho r^2 \delta x v$

    Therefore, using Newton's Second and Third Laws, the force that this mass of fluid exerts on the sphere is equal to the rate of change of its momentum; i.e.

    $\displaystyle \frac{k \rho r^2 \delta x v}{\delta t}$

    As $\displaystyle \delta t \rightarrow 0, \frac{\delta x}{\delta t}\rightarrow v$. So the force on the sphere is therefore

    $\displaystyle k \rho r^2v^2$

    Grandad
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