Velocity of a Spherical Particle in a Viscous Liquid: Integrating Drag Force

AI Thread Summary
The discussion focuses on deriving the x-component of velocity, vx(t), for a spherical particle moving through a viscous liquid, considering the drag force. The drag force is expressed as D = bv, where b is the drag coefficient calculated using the sphere's radius and the liquid's viscosity. Participants emphasize the need to integrate the acceleration function, which is dependent on velocity, rather than treating it as a constant. There is also a mention of the buoyant force, which should be considered alongside the drag force in the analysis. The conversation highlights the importance of applying Newton's second law and correctly integrating to find the velocity function.
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Homework Statement


An object moving in a liquid experiences a linear drag force: D⃗ =(bv, direction opposite the motion), where b is a constant called the drag coefficient. For a sphere of radius R, the drag constant can be computed as b=6πηR, where η is the viscosity of the liquid.

Find an algebraic expression for vx(t), the x-component of velocity as a function of time, for a spherical particle of radius R and mass m that is shot horizontally with initial speed v0 through a liquid of viscosity η.
Express your answer in terms of the variables v0, η, R, t, m, and appropriate constants.

Homework Equations





The Attempt at a Solution


Thinking of typical dynamics, I divided the drag force, bv by m to get the acceleration. Then I subtracted acceleration times time from the inivial velocity, v0. So it looked like this:

v0- (6πηRv0t)/m.

Obviously it wasn't right, as my teacher today told me that I have to integrate the acceleration function to get the velocity function. I have no idea how to integrate the acceleration function in which it looks like every single variable are constants.

Help would be appreciated! Thanks in advance.
 
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hi playoff! :smile:
playoff said:
… I have no idea how to integrate the acceleration function in which it looks like every single variable are constants.

no, a = dv/dt is a function of v, not a constant :wink:
 
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Draw a free body diagram, and apply Newton's second law to the mass. Don't forget to include the buoyant force.

Chet
 
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tiny-tim said:
hi playoff! :smile:


no, a = dv/dt is a function of v, not a constant :wink:

Ugh, I have a very shallow understanding in calculus. So if I would integrate it with v in the acceleration function, wouldn't it give me the position function in the velocity function? And the only variables I can use are v0, η, R, t, m, and appropriate constants.

Thanks for pointing it out though :D

@Chestermiller: I thought the only force acting in the x-axis is the drag force itself. Would the buoyant force also be acting against the velocity?
 
playoff said:
Ugh, I have a very shallow understanding in calculus. So if I would integrate it with v in the acceleration function, wouldn't it give me the position function in the velocity function? And the only variables I can use are v0, η, R, t, m, and appropriate constants.

Thanks for pointing it out though :D

@Chestermiller: I thought the only force acting in the x-axis is the drag force itself. Would the buoyant force also be acting against the velocity?
Oops. I should have read the problem statement more carefully. Sorry about that.

Chet
 
(just got up :zzz:)
playoff said:
… the position function in the velocity function?

i don't understand this :confused:

to integrate dv/dt = f(v),

write it dv/f(v) = dt, then integrate both sides :smile:
 
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