Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 12] Applying Definite Integrals to Solution of Physical Problems 177
1761**. Two electric charges * = 100 CGSE-and e l= =200 CGSE
lie on the x-axis at points * = and 1 *, cm, respectively.
What work will be done if the second charge is moved to point
* =10 cm?
2
1762**. A cylinder with a movable piston of diameter D = 20 cm
and length / = 80cm is filled with steam at a pressure
p=10kgfcm 2
. What
work must be done to halve the volume of
the steam with temperature kept constant (isoihermic process)?
1763**. Determine the work performed in the adiabatic expansion
of air (having initial volume u =l m 3
2
p _=l ft kgf/cm ) to volume u, = 10 m 8
?
1764**. A vertical shaft of weight P and
radius a rests on a bearing AB (Fig. 62).
The frictional force between a small part a
of the base of the shaft and the surface of
the support in contact with it is F==fipa,
where p = const is the pressure of the shaft
on the surface of the support referred to
unit area of the support, while pi is the coef-
ficient of friction. Find the work done by the
frictional force during one revolution of the
and pressure
shaft.
1765**. Calculate the kinetic energy
disk of mass M and radius R rotating
of a
with
angular velocity G> about an axis that passes through its centre
perpendicular to its plane.
1766. Calculate the kinetic energy of a right circular cone of
mass M rotating with angular velocity CD about its axis, if the
radius of the base of the cone is R and the altitude is H.
1767*. What work has to be don? to stop an iron sphere of
radius R = 2 me'res rotating with angular velocity w = 1,000 rpm
about its diameter? (Specific weight of iron, y = 7.8 s/cm j
.)
1768. A vertical triangle with base 6 and altitude h is sub-
merged vertex downwards in water so that its base is on the
surface of the water. Find the pressure of the water.
1769. A vertical dam has the shipa of a trapezoid. Calculate
the water pressure on the dam if we know that the upper base
a = 70 m, the lower base 6=50 m, and the height h = 20 m.
1770. Find the pressure of a liquid, whose specific weight is y.
on a vertical ellipse (with axes 2a and 26) whose centre is submerged
in the liquid to a distance h, while the major axis 2a
of the ellipse is parallel to the level of the liquid (h^b).
1771. Find the water pressure on a vertical circular cone
with radius of base R and altitude H submerged in walei vertex
downwards so that its base is on the surface of the water.