Problem Set #4
1. Euler’s First Law (or the Linear Momentum Principle, or Conservation of Linear Momentum) is
sometimes written in words as
The rate of change of
momentum of a fluid particle
The sum of the forces
acting on the particle
In a static fluid, the rate of change of momentum is zero, and the forces acting on a fluid particle are
generally the gravitational force and pressure, as described in lecture.
We showed therefore that, in a
is the equation that determines the pressure distribution.
Consider a case where the static fluid is in a vehicle that undergoing a constant acceleration a
rate of change of momentum of a fluid particle with mass m is then ma
, or for an infinitesimally small
particle with volume dV, it would be
Derive the equation for the pressure distribution in a fluid
undergoing constant acceleration a
Now consider a cylindrical glass of water with radius R on a horizontal turntable, rotating about its
axis of symmetry (the z-axis, where radial position r=0).
In the absence of rotation, the height of water
in the glass is H.
As you learned in physics, each position in an object undergoing pure rotation is
accelerating inward, with centripetal acceleration a
where r is the distance from the axis of rotation in a cylindrical coordinate system,
is the rotational
velocity, and e
is a unit vector pointing away from the z-axis.
Use this expression for centripetal
acceleration with your result from part “a” to derive an expression for the height h(r) of the air-water
interface in a rotating, cylindrical glass of water.