- Let R denote the finite region bounded by z = 4 − x
^{2}− y^{2}and the xy plane. Find z_{c}, the z coordinate of the center of mass if the density σ is a constant. - Let R denote the finite region bounded by z = 4 − x
^{2}− y^{2}and the xy plane. Find z_{c}, the z coordinate of the center of mass if the density σ is equals σ= z. - Find the mass and center of mass of the region between the surfaces z = −y
^{2}+ 8 and z = 2x^{2}+ y^{2}if the density equals σ = 1. - Find the mass and center of mass of the region between the surfaces z = −y
^{2}+ 8 and z = 2x^{2}+ y^{2}if the density equals σ= x^{2}. - The two cylinders, x
^{2}+ y^{2}= 4 and y^{2}+ z^{2}= 4 intersect in a region R. Find the mass and center of mass if the density σ, is given by σ= z^{2}. - The two cylinders, x
^{2}+ y^{2}= 4 and y^{2}+ z^{2}= 4 intersect in a region R. Find the mass and center of mass if the density σ, is given by σ= 4 + z. - Find the mass and center of mass of the set such that++ z
^{2}≤ 1 if the density is σ= 4 + y + z. - Let R denote the finite region bounded by z = 9−x
^{2}−y^{2}and the xy plane. Find the moment of inertia of this shape about the z axis given the density equals 1. - Let R denote the finite region bounded by z = 9−x
^{2}−y^{2}and the xy plane. Find the moment of inertia of this shape about the x axis given the density equals 1. - Let B be a solid ball of constant density and radius R. Find the moment of inertia about a
line through a diameter of the ball. You should get R
^{2}M where M is the mass.. - Let B be a solid ball of density σ = ρ where ρ is the distance to the center of the ball which has radius R. Find the moment of inertia about a line through a diameter of the ball. Write your answer in terms of the total mass and the radius as was done in the constant density case.
- Let C be a solid cylinder of constant density and radius R. Find the moment of inertia about
the axis of the cylinder
You should get

R^{2}M where M is the mass. - Let C be a solid cylinder of constant density and radius R and mass M and let B be a solid ball of radius R and mass M. The cylinder and the ball are placed on the top of an inclined plane and allowed to roll to the bottom. Which one will arrive first and why?
- A ball of radius 4 has a cone taken out of the top which has an angle of π∕2 and then a cone taken out of the bottom which has an angle of π∕3. If the density is λ = ρ, find the z component of the center of mass.
- A ball of radius 4 has a cone taken out of the top which has an angle of π∕2 and then a cone taken out of the bottom which has an angle of π∕3. If the density is λ = ρ, find the moment of inertia about the z axis.
- Suppose a solid of mass M occupying the region B has moment of inertia, I
_{l}about a line, l which passes through the center of mass of M and let l_{1}be another line parallel to l and at a distance of a from l. Then the parallel axis theorem states I_{l1}= I_{l}+ a^{2}M. Prove the parallel axis theorem. Hint: Choose axes such that the z axis is l and l_{1}passes through the pointin the xy plane. ^{∗}Using the parallel axis theorem find the moment of inertia of a solid ball of radius R and mass M about an axis located at a distance of a from the center of the ball. Your answer should be Ma^{2}+MR^{2}.- Consider all axes in computing the moment of inertia of a solid. Will the smallest possible moment of inertia always result from using an axis which goes through the center of mass?
- Find the moment of inertia of a solid thin rod of length l, mass M, and constant density about
an axis through the center of the rod perpendicular to the axis of the rod. You should get
l
^{2}M. - Using the parallel axis theorem, find the moment of inertia of a solid thin rod of length l, mass
M, and constant density about an axis through an end of the rod perpendicular to the axis of
the rod. You should get l
^{2}M. - Let the angle between the z axis and the sides of a right circular cone be α. Also assume the height of this cone is h. Find the z coordinate of the center of mass of this cone in terms of α and h assuming the density is constant.
- Let the angle between the z axis and the sides of a right circular cone be α. Also assume the height of this cone is h. Assuming the density is σ = 1, find the moment of inertia about the z axis in terms of α and h.
- Let R denote the part of the solid ball, x
^{2}+y^{2}+z^{2}≤ R^{2}which lies in the first octant. That is x,y,z ≥ 0. Find the coordinates of the center of mass if the density is constant. Your answer for one of the coordinates for the center of mass should beR. - Show that in general for L angular momentum,
where Γ is the total torque,

where F

_{i}is the force on the i^{th}point mass.

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