A solid S occupies the region of space located outside the
sphere x2 + y2 +...
A solid S occupies the region of space located outside the
sphere x2 + y2 + z2 = 8 and inside
the sphere x2 + y2 + (z - 2)2 = 4.
The density of this solid is proportional to the distance from the
origin.
Determine the center of mass of S.
Is the center of mass located inside the solid S ?
Carefully justify your answer.
Consider the unit sphere x2 +y2 +z2 = 1 and the cone (z+√2)2 =
x2 +y2. Show that these surfaces are tangent where they intersect,
that is, for a point on the intersection, these surfaces have the
same tangent plane
Let D be the region bounded by the paraboloids z = 8 -
x2 - y2 and z = x2 +
y2. Write six different triple iterated integrals for
the volume of D. Evaluate one of the integrals.
A space station is located in a gravity-free region of
space. It consists of a large diameter, hollow thin-walled cylinder
which is rotating freely about its axis. It is spinning at a speed
such that the apparent gravity on the inner surface is the same as
that on earth. The cylinder is of radius r and mass M. (a) What is
the minimum total work which had to be done to get the cylinder
spinning up to speed. (b) Radial...
A lamina occupies the region inside the circle x^2 + y^2 = 6x,
but outside the circle x^2 + y^2 =9.
Find the center of mass if the density at any point is inversely
proportional to its distance from the origin.
A lamina occupies the region inside the circle x^2 + y^2 = 6x,
but outside the circle x^2 + y^2 =9.
Find the center of mass if the density at any point is inversely
proportional to its distance from the origin.
Evaluate the following integral,
∫
∫
S
(x2 + y2 + z2) dS,
where S is the part of the cylinder x2 +
y2 = 64 between the planes z = 0 and
z = 7, together with its top and bottom disks.
The magnetic flux density in a region of free space is given by
B = −x2 ax + 2y3 ay − 2z az T. Find the total
force on the rectangular loop shown which lies in the plane y = 0
and is bounded by x = 1, x = 3, Z= 2, and Z= 5, all dimensions in
cm. The current is 30 ampere in anticlockwise direction.