Question

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.

Answer #1

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 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.

Find the moment of inertia of a uniform hollow sphere of mass M,
inner radius r, and outer radius R > r, about an axis through
the center of mass. Consider your answer for the cases r → 0 and r
→ R. Does the result reduce correctly? Explain.

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).

You place a sphere of radius R=1.54 m and of density ρ =
0.75g/cm3 in a pool of clean water. The sphere begins to float and
a portion of the sphere begins to show. (a) What is the height of
the portion that shows above the water? What minimum force should
you press on the sphere to make it sink completely in the pool?
Show your complete work step by step.

A sphere with a radius of 0.232 m has a uniform charge density
and a total charge of 70.5 mC. What is the magnitude of the
electric field at each of the following locations?
(a) a distance of 0.150 m from the center
N/C
(b) a distance of 0.232 m from the center
N/C
(c) a distance of 0.550 m from the center
N/C

You place a sphere of radius R=1.54 m and of density ρ =
0.75g/cm^3 in a pool of clean water. The sphere begins to float and
a portion of the sphere begins to show. What is the height of the
portion that shows above the water? What minimum force should you
press on the sphere to make it sink completely in the pool? Show
your complete work step by step.

The density of a cylinder of radius R and length l
varies linearly from the central axis where ρ1=500
kg/m3 to the value ρ2=3ρ1. If
R=.05 m and l= .1 m, find:
a. The average density of the cylinder over the radius.
b. The average density over its volume.
c. the moment of inertia of the cylinder about its central
axis.

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.

A sphere with radius R carries a positive charge density of
ρ=2r3, take the potential at infinity is 0,
find the electric potential everywhere (r<R and r>R).

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