Consider a solid hemisphere of radius R, constant mass density ρ, and a total mass M....
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.
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Consider a solid hemisphere of radius R, constant mass density
ρ, and a total mass M....
1. Consider a solid cone with uniform density p, height h, and circular base with radius R (hence mass M,sphR2). Let the vertex of the cone be the origin ofthe body frame. By symmetry, choose bas...
1. Consider a solid cone with uniform density p, height h, and circular base with radius R (hence mass M,sphR2). Let the vertex of the cone be the origin ofthe body frame. By symmetry, choose basis vectors e for the body frame such that the inertia tensor I, is diagonal. Will this rigid body with this body origin be an asymmetric top, a symmetric top, or a spherical top? Calculate the inertia tensor in this basis How will the inertia...
Imagine a hypothetical star of radius R, whose mass density ρ is constant throughout the star....
Imagine a hypothetical star of radius R, whose mass density ρ is constant throughout the star. The star is composed of a classical ideal gas of ionized hydrogen, so there are free protons and free electrons flying around providing the pressure support. The star is in hydrostatic equilibrium (a) What is the pressure as a function of radius in the star, P(r)? As a boundary condition, the pressure at the surface should be zero, P(R) 0 (b) What is the...
Consider a cylinder of mass M, radius R and length L. (a) Calculate the inertia tensor...
Consider a cylinder of mass M, radius R and length L. (a) Calculate the inertia tensor for rotations about the center of mass in the frame where the z axis is along the axis of the cylinder. Use cylindrical coordinates, where x = r cos θ and y = r sin θ. (b) Find the inertia tensor in the frame where the center of the “bottom side” is at the origin with the z axis along the axis of the...
A solid cylinder of height L and radius R has uniform mass density . Find the...
A solid cylinder of height L and radius R has uniform mass density . Find the moment of inertia tensor about the center of the cylinder. For what value of L/R is the cylinder equally easy to spin about any axis?
Consider a solid sphere of radius R = 0.6 m that is uniformly charged with ρ...
Consider a solid sphere of radius R = 0.6 m that is uniformly charged with ρ = -16 μC/m3. What is the electric potential a distance 2 m from the center of the sphere?
Consider a solid sphere of radius R = 0.8 m that is uniformly charged with ρ...
Consider a solid sphere of radius R = 0.8 m that is uniformly charged with ρ = -10 μC/m3. What is the electric potential a distance 2 m from the center of the sphere?
Consider a solid sphere of radius R = 0.6 m that is uniformly charged with ρ...
Consider a solid sphere of radius R = 0.6 m that is uniformly charged with ρ = -10 μC/m3. What is the electric potential a distance 3 m from the center of the sphere?
Find the center of mass of a hemisphere with radius r, whose density is uniform.
Find the center of mass of a hemisphere with radius r, whose density is uniform.
3rd Question Consider a solid insulating sphere of radius b with nonuniform charge density ρ-ar, where...
3rd Question
Consider a solid insulating sphere of radius b with nonuniform charge density ρ-ar, where a is a constant. Find the charge contained within the radius r< bas in the figure. The volume element dV for a spherical shell of radius r and thickness dr is equal to 4 π r2 dr.
1. Consider a solid cone with uniform density p, height h, and circular base with radius R (hence mass M,sphR2). Let the vertex of the cone be the origin ofthe body frame. By symmetry, choose basis vectors e for the body frame such that the inertia tensor I, is diagonal. Will this rigid body with this body origin be an asymmetric top, a symmetric top, or a spherical top? Calculate the inertia tensor in this basis How will the inertia...
Imagine a hypothetical star of radius R, whose mass density ρ is constant throughout the star. The star is composed of a classical ideal gas of ionized hydrogen, so there are free protons and free electrons flying around providing the pressure support. The star is in hydrostatic equilibrium (a) What is the pressure as a function of radius in the star, P(r)? As a boundary condition, the pressure at the surface should be zero, P(R) 0 (b) What is the...
Find the center of mass of a hemisphere with radius r, whose density is uniform.
3rd Question
Consider a solid insulating sphere of radius b with nonuniform charge density ρ-ar, where a is a constant. Find the charge contained within the radius r< bas in the figure. The volume element dV for a spherical shell of radius r and thickness dr is equal to 4 π r2 dr.
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