GEF 2500 Problem set 1
1. Solve the following:
(a)
du
dt
(b)
d2 u
dt2
(c)
+ A = 0,
u(t = 0) = B,
A, B = const
+ A = 0,
u(t = 0) = B,
du
dt ( t
dP
dt
= kP,
P(t = 0) = 1000,
(d)
d2 u
dt2
+ ku = 0,
(e)
d2 f
dx2
= x,
u ( t = 0) = u 0 ,
f ( x = 0) = 0,
= 0) = 0,
A, B = const
P(1) = 2000,
du
dt ( t
df
dx ( x
= 0) = 0,
= x0 ) = F0 ,
k = const
u0 = const
F0 , x0 = const
(f) Consider a domain 0 ≤ x ≤ L
Calculate:
d2ψ
dx2
+ βr ddxψ = − A,
ψ( x = 0) = 0, ψ( x = L) = 0
(Hint: multiply through by an integrating factor eβ x/r and assume βrL large)
2. Water of density ρ flows out of a circular tap of cross-sectional area A
at velocity V0 at height z=0
(a) Treating the water as incompressible, and neglecting vertical pressure forces within the water by comparison with the downward
force due to gravity, calculate the velocity V ( z) as it descends
from the tap as a function of the distance z below the tap (note
that here z is directed downward!)
(b) Considering mass conservation and your solution to (a), find the
water jet radius R(z).
1
(c) By considering the lateral acceleration of the water on the edge
of the water jet implied by your solution in (a), infer where the
water pressure inside the water jet is higher vs. lower than the
ambient air pressure around the water jet.
3. Consider a hollow circular glass pipe (see figure). The circle into which
the pipe is bent is 1m in radius and the inner radius of the pipe 1 cm.
The upper half of the pipe contains a vacuum and its lower half contains water at rest of density ρ=1000kg m−3
(a) Roughly what is the water pressure at the bottom of the circular
pipe?
(b) The air surrounding the pipe is at room temperature of 20◦ C and
pressure 1000hPa. If a small hole is drilled in the bottom of the
circular pipe, will water flow out of the pipe, will air flow into
the pipe, or will nothing happen?
(c) How small hole can we drill without having any water flowing
out of the pipe or air flowing into the pipe due to the effect of
surface tension? The relationship between pressure and surface
tension is ∆P = 2Rσ , where σ is the surface tension and R is the
radius of the hole.
(Hint: the surface tension at 20◦ C is σ =0.0728 Nm−1 )
vacuum
air
water
Hole location for (b)
Figur 1: Problem 3
2