Question

3. Consider a solid hemisphere of radius R, constant mass density ρ, and a total mass M. Calculate all elements of the inertia tensor (in terms of M and R) of the hemisphere for a reference frame with its origin at the center of the circular base of the hemisphere. Make sure to clearly sketch the hemisphere and axes positions.

Answer #1

Use spherical coordinates.
(a) Find the centroid of a solid homogeneous hemisphere of
radius 1. (Assume the upper hemisphere of a sphere centered at the
origin. Use the density function
ρ(x, y,
z) = K.
(x, y, z) =
(b) Find the moment of inertia of the solid in part (a) about a
diameter of its base.
Id =

A sphere of radius R has total mass M and density function given
by ρ = kr, where r is the distance a point lies from the centre of
the sphere. Give an expression for the constant k in terms of M and
R.

A solid sphere, radius R, is centered at the origin. The
“northern” hemisphere carries a uniform charge density ρ0, and the
“southern” hemisphere a uniform charge density −ρ0. Find the
approximate field E(r,θ) for points far from the sphere (r ≫
R).

A sphere of solid aluminum (r = 2.7 g/cm3 ) has a radius R, a
mass M, and a moment of inertia I0 about its center. A second
sphere of solid aluminum has a different mass M’ with a radius 2R.
What is the moment of inertia of the second sphere about its center
I in terms of the first sphere I0? Mass should not appear in your
answer (4 points).

A sphere with radius 0.150 m has density ρ that decreases with
distance r from the center of the sphere according to
ρ=3.25×103kg/m3−(8.50×103kg/m^4)r .
a)Calculate the total mass of the sphere. Express your answer
with the appropriate units.
B)Calculate the moment of inertia of the sphere for an axis
along a diameter. Express your answer with the appropriate
units.

A sphere with radius 0.250 m has density ρ that decreases with
distance r from the center of the sphere according to
ρ=2.75×103kg/m3−(9.00×103kg/m4)r . Part A: Calculate the total mass
of the sphere. Express your answer with the appropriate units. M =
?; Try Again Part B :Calculate the moment of inertia of the sphere
for an axis along a diameter. Express your answer with the
appropriate units.

A small solid sphere of mass M0, of radius
R0, and of uniform density ρ0 is placed in a
large bowl containing water. It floats and the level of the water
in the dish is L. Given the information below, determine the
possible effects on the water level L, (R-Rises, F-Falls,
U-Unchanged), when that sphere is replaced by a new solid sphere of
uniform density.
(a) The new sphere has density ρ = ρ0 and mass M >
M0
(b)...

A small solid sphere of mass M0, of radius
R0, and of uniform density ρ0 is placed in a
large bowl containing water. It floats and the level of the water
in the dish is L. Given the information below, determine the
possible effects on the water level L, (R-Rises, F-Falls,
U-Unchanged), when that sphere is replaced by a new solid sphere of
uniform density.
Options: R F U R or U F or U R or F or U
The...

A small solid sphere of mass M0, of radius
R0, and of uniform density ρ0 is placed in a
large bowl containing water. It floats and the level of the water
in the dish is L. Given the information below, determine the
possible effects on the water level L, (R-Rises, F-Falls,
U-Unchanged), when that sphere is replaced by a new solid sphere of
uniform density.
Options are:
R= Rises
F= Falls
U= Unchnaged
F or U
R or U
F...

A small solid sphere of mass M0, of radius
R0, and of uniform density ρ0 is placed in a
large bowl containing water. It floats and the level of the water
in the dish is L. Given the information below, determine the
possible effects on the water level L, (R-Rises, F-Falls,
U-Unchanged), when that sphere is replaced by a new solid sphere of
uniform density.
Read it to me
The new sphere has radius R > R0 and density ρ...

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