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Fluid Mechanics: Problems and Solutions, Study Guides, Projects, Research of Physics

irodov_problems_in_general_physics_2011

Typology: Study Guides, Projects, Research

2010/2011

Uploaded on 01/07/2023

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fice to
s,
with
s
being considerably less than the piston area. The
friction and viscosity are negligibly small.
1.323. A cylindrical vessel of height
h
and base area
S
is filled
with water. An orifice of area
s
<
S
is opened in
vessel. Neglecting the
๎˜‰
viscosity
๎˜‰
of wa-
ter, determine how soon all
๎˜‰
the water
will pour out of the vessel.
1.324.
๎˜‰
A
๎˜‰
horizontally
๎˜‰
oriented
๎˜‰
tube
the bottom of the
1.1!
h
AB
of length
1
rotates with a constant
angular
๎˜‰
velocity
๎˜‰
co
๎˜‰
about a stationary
II
vertical axis
00'
passing through the end
ho
A
(Fig. 1.86). The tube is filled with an
ideal fluid. The
๎˜‰
A
๎˜‰
the tube is
---
end
๎˜‰
of
๎˜‰
open,
the closed end
B
has a very small orifice.
Find the velocity of the fluid relative to
the
๎˜‰
tube
๎˜‰
as
๎˜‰
a
๎˜‰
function of the column
v _
"height"
h.
1.325.
๎˜‰
Demonstrate that in
๎˜‰
the
๎˜‰
case
of a steady
๎˜‰
flow
๎˜‰
of
๎˜‰
an
๎˜‰
ideal fluid Eq.
๎˜‰
Fig. 1.83.
(1.7a) turns into Bernoulli equation.
1.326. On the opposite sides of a wide vertical vessel filled with
water two identical holes are opened, each having the cross-sectional
Fig. 1.84. Fig. 1.85.
area
S =
0.50 cm
2
. The height difference between them is equal to
Ah = 51 cm. Find the resultant force of reaction of the water flow-
ing out of the vessel.
1.327. The side wall of a wide vertical cylindrical vessel of height
h =
75 cm has a narrow vertical slit running all the way down to
the bottom of the vessel. The length of the slit is
1 =
50 cm and the
width
b =
1.0 mm. With the slit closed, the vessel is filled with
water. Find the resultant force of reaction of the water flowing out of
the vessel immediately after the slit is opened.
1.328. Water flows out of a big tank along a tube bent at right an-
gles: the inside radius of the tube is equal to
r =
0.50 cm (Fig. 1.87).
The length of the horizontal section of the tube is equal to
1 =
22 cm.
The water flow rate is
Q =
0.50 litres per second. Find the moment
of reaction forces of flowing water, acting on the tube's walls, relative
to the point
0.
64
pf3
pf4
pf5

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fice to s, with s being considerably less than the piston area. The

friction and viscosity are negligibly small.

1.323. A cylindrical vessel of height h and base area^ S^ is filled

with water. An orifice of area s < S is opened in vessel. Neglecting the viscosity of wa- ter, determine how soon all the water will pour out of the vessel. 1.324. A horizontally oriented tube

the bottom of the

1.1!

h

AB of length 1 rotates with a constant

angular velocity co about a stationary

II

vertical axis 00' passing through the end ho

A (Fig. 1.86). The tube is filled with an

ideal fluid. The end A of the tube is open, ---

the closed end B has a very small orifice.

Find the velocity of the fluid relative to the tube as a function of the column

v _

"height" h.

1.325. Demonstrate that in the case of a steady flow of an ideal fluid Eq. Fig. 1.83. (1.7a) turns into Bernoulli equation. 1.326. On the opposite sides of a wide vertical vessel filled with water two identical holes are opened, each having the cross-sectional

Fig. 1.84. Fig. 1.85.

area S = 0.50 cm2. The height difference between them is equal to

Ah = 51 cm. Find the resultant force of reaction of the water flow- ing out of the vessel. 1.327. The side wall of a wide vertical cylindrical vessel of height

h = 75 cm has a narrow vertical slit running all the way down to

the bottom of the vessel. The length of the slit is 1 = 50 cm and the

width b = 1.0 mm. With the slit closed, the vessel is filled with

water. Find the resultant force of reaction of the water flowing out of the vessel immediately after the slit is opened. 1.328. Water flows out of a big tank along a tube bent at right an-

gles: the inside radius of the tube is equal to r = 0.50 cm (Fig. 1.87).

The length of the horizontal section of the tube is equal to 1 = 22 cm.

The water flow rate is Q = 0.50 litres per second. Find the moment

of reaction forces of flowing water, acting on the tube's walls, relative

to the point 0.

64

C Z

Fig. 1.87.

Fig. 1.89.

1.329. A side wall of a wide open tank is provided with a narrow-

ing tube (Fig. 1.88) through which water flows out. The cross-sectional

area of the tube decreases from S = 3.0 cm2 to s = 1.0 cm2. The

water level in the tank is h = 4.6 m higher than that in the tube.

W

Fig. 1.86.

Neglecting the viscosity of the water, find the horizontal component

of the force tending to pull the tube out of the tank.

Fig. 1.88.

1.330. A cylindrical vessel with water is rotated about its ver-

tical axis with a constant angular velocity co. Find:

(a) the shape of the free surface of the water;

(b) the water pressure distribution over the bottom of the vessel

along its radius provided the pressure at the central point is equal to

Po.

1.331. A thin horizontal disc of radius R =^ 10 cm is located with-

in a cylindrical cavity filled with oil whose viscosity it (^) = 0.08 P

(Fig. 1.89). The clearance between the disc and the horizontal planes

of the cavity is equal to h = 1.0 mm. Find the power developed by

the viscous forces acting on the disc when it rotates with the angular

velocity co = 60 rad/s. The end effects are to be neglected.

5-9451 (^65)

viscosities of glycerin and water are equal to rh = 13.9 P and 12 = = 0.011 P respectively. 1.338. A lead sphere is steadily sinking in glycerin whose viscosity is equal toil = 13.9 P. What is the maximum diameter of the sphere at which the flow around that sphere still remains laminar? It is known that the transition to the turbulent flow corresponds to Rey- nolds number Re = 0.5. (Here the characteristic length is taken to be the sphere diameter.) 1.339. A steel ball of diameter d = 3.0 mm starts sinking with zero initial velocity in olive oil whose viscosity is 1 = 0.90 P. How soon after the beginning of motion will the velocity of the ball differ from the steady-state velocity by n = 1.0%?

1.8. RELATIVISTIC MECHANICS

1 โ€” (v/c) where /0is the proper length and Atois the proper time of the moving clock.

  • Lorentz transformation*: xโ€”Vt t 1

โ€” XVIC 2 โ€” (^) e โ€” (V/c)2 โ€”^ (V/c)2^ โ€ข^

(1.8b)

  • Interval 812is an invariant:

s2 12 =c2t22 _1q2 f-= in , (I .8c)

where t12is the time interval between events 1 and 2, /12is the distance between the points at which these events occurred.

  • Transformation of velocity*:
    • (1.8d)

(1.8e)

(1.8f)

(1.8g)

' vsโ€”^ V^ vy"1/-1 โ€”^ (V/c) voc 1 โ€”vxV/c2 '

v, 1 โ€”vxV/c

  • Relativistic mass and relativistic momentum:

M โ€” mo^ p= my^ = MoV y1โ€” (v/02^ y^ 1โ€” (v/ where mois the rest mass, or, simply, the mass.

  • Relativistic equation of dynamics for a particle: dp F dt where p is the relativistic momentum of the particle.
  • Total and kinetic energies of a relativistic particle: E (^) = mc2 = m0,2 + (^) T, T -= (m โ€” mo ) c 2.
  • The reference frame K' is assumed to move with a velocity V in the posi- tive direction of the x axis of the frame K, with the x' and^ x^ axes coinciding and the y' and y axes parallel.
  • Lorentz contraction of length and slowing of a moving chick:

/ = /0 -V 1 โ€”(v/c)2 , At= A to^ (1.8a)

5* 67

  • Relationship between the energy and momentum of a relativistic par- tide

E2 โ€” p2c2= mic4, pc = Y T (T ยฑ2moc4). (1.8h)

  • When considering the collisions of particles it helps to use the follow- ing invariant quantity: E2 p2c2 = mic4, (^) (1.81)

where E and p are the total energy and momentum of the system prior to the collision, and mois the rest mass of the particle (or the system) formed.

1.340. A rod moves lengthwise with a constant velocity v relative

to the inertial reference frame K. At what value of v will the length

of the rod in this frame be 1 = 0.5% less than its proper length?

1.341. In a triangle the proper length of each side equals a. Find

the perimeter of this triangle in the reference frame moving relative

to it with a constant velocity V along one of its

(a) bisectors; (b) sides.

Investigate the results obtained at V <c and V -4- c, where c is the

velocity of light.

1.342. Find the proper length of a rod if in the laboratory frame

of reference its velocity is v = c/2, the length 1 = 1.00 m, and the

angle between the rod and its direction of motion is 0 = 45ยฐ.

1.343. A stationary upright cone has a taper angle 0 =. 45ยฐ,

and the area of the lateral surface So= 4.0 m2. Find: (a) its

taper angle; (b) its lateral surface area, in the reference frame

moving with a velocity v = (4/5)c along the axis of the cone.

1.344. With what velocity (relative to the reference frame K) did

the clock move, if during the time interval t = 5.0 s, measured by

the clock of the frame K, it became slow by At = 0.10 s?

1.345. A rod flies with constant velocity past a mark which is

stationary in the reference frame K. In the frame K it takes At

20 ns for the rod to fly past the mark. In the reference frame fixed

to the rod the mark moves past the rod for At' = 25 ns. Find the prop-

er length of the rod.

1.346. The proper lifetime of an unstable particle is equal to

Ato= 10 ns. Find the distance this particle will traverse till its

decay in the laboratory fraine of reference, where its lifetime is equal

to At = 20 ns.

1.347. In the reference frame K a muon moving with a velocity

v = 0.990c travelled a distance 1 = 3.0 km from its birthplace to

the point where it decayed. Find:

(a) the proper lifetime of this muon;

(b) the distance travelled by the muon in the frame K "from the

muon's standpoint".

1.348. Two particles moving in a laboratory frame of reference

along the same straight line with the same velocity v = (314)c strike

against a stationary target with the time interval At = 50 ns. Find