Drag force on a descending sphere

[Est120]

Homework Statement

compute the drag force on the sphere in dynes

Relevant Equations

weight of fluid displaced= rho*4/3pi*R^3
weight of hollow sphere= mg

i tried a force balance around the sphere but the weight of the displaced fluid is greater than the weight of the sphere which gives a net acceleration upwards and no terminal velocity but the book says that the terminal velocity has a certain value from there the exercise is meaningless to me

i Attach a photo of the solution where the numerical answer comes from, but as I told you, I no longer understood why a terminal velocity is reached if the Archimedean force is always constant and the weight of the sphere is also constant , so adding a force due to the resistance around the liquid would only accelerate the sphere upwards and it would never reach a constant terminal velocity in downwards direction. the book is Bird's Transporth phenomena page 193

 

[TSny]

...adding a force due to the resistance around the liquid would only accelerate the sphere upwards and it would never reach a constant terminal velocity in downwards direction.

Are you assuming that the drag force must act in the upward direction? If so, why?

 

[Lnewqban]

The downwards resistive force of drag depends on the value of the increasing upwards velocity; hence, a balance of forces can be reached at some velocity.

Please, see:
https://en.m.wikipedia.org/wiki/Drag_(physics)

 

[Est120]

Are you assuming that the drag force must act in the upward direction? If so, why?

I believed that the force was going against the movement of the sphere and now it turns out that it is going downwards "helping" the sphere to fall

 

[haruspex]

helping" the sphere to fall

How do you know it is falling?

 

[Est120]

I think I understand now, the sphere at the beginning goes up and the friction always pushes it downwards but it is not enough, only when it acquires enough speed it reaches the terminal speed

 

[haruspex]

I think I understand now, the sphere at the beginning goes up and the friction always pushes it downwards but it is not enough, only when it acquires enough speed it reaches the terminal speed

Yes.