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

344_Differential Equations_[Ch. 9
2898*. Prove that for a heavy liquid rotating about a vertical
axis the free surface has the form of a paraboloid of revolution.
2899*. Find the relationship between the air pressure and the
altitude if it is known that the pressure is 1 kgf on 1 cm 2
at
sea level and 0.92 kgf on 1 cm 2
at an altitude of 500 metres.
2900*. According to Hooke's law an elastic band of length
/ increases in length klF(k = const) due to a tensile force F.
By how much will the band increase in length due to its weight
W if the band is suspended at one end? (The initial length of
the band is /.)
2901. Solve the same problem for a weight P suspended from
the end of the band.
When solving Problems 2902 and 2903, make use of Newton's
to the
law, by which the rate of cooling of a body is proportional
difference of temperatures of the body and the ambient medium.
2902. Find the relationship between the temperature T and
the time f if a body, heated to T degrees, is brought into a room
at constant temperature (a degrees).
2903. During what time will a body
heated to 100 cool off
to 30 if the temperature of the room is 20 and during the first
20 minutes the body cooled to 60?
2904. The retarding action of friction on a disk rotating in
of rotation. Find
a liquid is proportional to the angular velocity
and time if it is
the relationship between the angular velocity
known that the disk began rotating at 100 rpm and after one
minute was rotating at 60 rpm.
2905*. The rate of disintegration of radium is proportional
to the quantity of radium present. Radium disintegrates by one
half in 1600 years. Find the percentage of radium that has disinte-
grated after 100 years.
2906*. The rate of outflow of water from an aperture at
a vertical distance h from the free surface is defined by the
formula
where c0.6 and g is
During what period
the
of
acceleration of
time will the
gravity.
water filling a hemispherical
boiler of diameter 2 metres flow out of it through a circular
opening of radius 0.1 m in the bottom.
2907*. The quantity of light absorbed in passing through
a thin layer of water is proportional to the quantity of incident
light and to the thickness of the layer. If one half of the original
quantity of light is absorbed in passing through a three-metrethick
layer of water, what part of this quantity will reach a depth
of 30 metres?