# Object in a test tube

#### Swatch

An object is put in a test tube . The test tube is put in a centrifugal machine where it spins around. Would it be correct to say that the net force on the object is zero since it is not moving. I know the test tube has a net inward force that makes it change direction. But the object wants to fly out of the tube but is stopped by the bottom of the test tube.

Am I totally wrong here?

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#### Swatch

What I meant is that the object doesn't accelerate. But then again it changes direction. A direct answer would be appreciated.

#### Doc Al

Mentor
centripetal acceleration

An object spinning in a circle is most definitely accelerating! It is accelerating towards the center of the circle; this is usually called centripetal acceleration. And, since it is accelerating, there must be a net force pulling the object towards the center.

#### Swatch

That's what I thought. But wasn't sure because the object isn't in an inertial frame of reference.

#### MalleusScientiarum

If an object isn't in an inertial frame of reference then pretty much by definition you will observe that a force is acting on it.

#### Swatch

O.K. I think I understand this but the real problem I don't understand, here it comes.

An incompressible fluid with density rho is in a horizontal test tube of inner cross-sectional area A. The test tube spins in a horizontal circle with angular speed w. Gravitational forces are negligeble. An object of volume V and density RHOob has its center of mass at a distance Rcmob from the axis. Show that the net horizontal force on the object is rhoVw^2Rcm, where Rcm is the distance from the axis to the center of mass of the displaced fluid. I have already derived expressions for the pressure at a distance from the surface of the water, and for the pressure difference for a volume element of thickness dr. It seems to me that the net force on the object is the mass*acceleration of the displaced fluid volume.

I don't know how to treat this problem. I see the force as the mass*acceleration of the object + displaced water.

#### Doc Al

Mentor
Swatch said:
It seems to me that the net force on the object is the mass*acceleration of the displaced fluid volume.
The net force on any object is its mass time its acceleration. This object is being centripetally accelerated, so $F_{net} = m a_c$.

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