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What happens when you swing an object that is attached to a string?
The object does not move in a straight line but instead moves in a
circular orbit. So what stops the object moving in a straight line as
predicted by Newton? The answer is the force due to the tension in the
string which accelerates the object. We can see this in Figure
2.4.
Figure 2.4:
If the radius is fixed as , then the velocity changes
direction as the object rotates. Hence, there is a change in velocity
and so an acceleration.

The speed of the object is , where
is the angle. The velocity is
and there is a change in the direction of the velocity with time
(e.g. when ,
but when ,
). Since there is a change in the
velocity, there must be an acceleration and it is in the
negative radial direction.
The acceleration is due to the tension in the string where
For a uniform, imcompressible liquid in a cylindrical can rotating steadily
about the
axis, what is the equation of its surface? Set up cylindrical
polars. Figure 2.5 illustrates the situation.
Figure 2.5:
Forces acting on a fluid in a rotating cylindrical can.

The basic equation is

(2.14) 
If the velocity is only in the direction, then (2.14)
reduces to
Thus, considering the radial and vertical direction separately we have
and
is constant (the can is rotating at a constant rate).
Integrating the radial component gives
where is an arbitrary function obtained by integrating with
respect to . Substituting into the vertical component gives
and so
At the surface of the liquid the pressure, , is constant and so the
equation of the surface is (, )

(2.15) 
The equation is a parabola. Note that as
,
and that as increases the slope increases.
Next: General Properties of a
Up: Fluids and Nonmagnetic Plasmas
Previous: Hydrostatic Liquids and Atmospheres
Prof. Alan Hood
20001106