Mar
29
Moment of Inertia Problem?
The methane molecule (CH4) has four hydrogen atoms located at the vertices of a regular tetrahedron of edge length 0.18 nm, with the carbon atom at the center of the tetrahedron. Find the moment of inertia (in kg * m^2) of this molecule for rotation about an axis that passes through the centers of the carbon atom and one of the hydrogen atoms.
Can somebody help me with this? I keep getting the wrong answer.
Thanks!
The methane molecule (CH4) has four hydrogen atoms located at the vertices of a regular tetrahedron of edge length 0.18 nm, with the carbon atom at the center of the tetrahedron. Find the moment of inertia (in kg * m^2) of this molecule for rotation about an axis that passes through the centers of the carbon atom and one of the hydrogen atoms.
Each face of a regular tetrahedron is an equilateral triangle.
I drew an equilateral triangle.
The base = length of edge
The altitude = (length of edge) * sin 60°.
The axis of rotation is the vertical line from the top point of the tetrahedron to the centroid of the equilateral triangle that is the base of the tetrahedron.
The distance from the centroid of the equilateral triangle that is the base of the tetrahedron to the top point of the tetrahedron = (length of edge) * sin 60°.
This is the altitude of the tetrahedron.
The distance from the centroid of the equilateral triangle that is the base of the tetrahedron up to the perpendicular line from centroid of the equilateral triangle that is a face of the tetrahedron = [(length of edge) ÷ 2] * tan 30°.
The perpendicular distance from this point to centroid of the equilateral triangle that is a face of the tetrahedron = [(length of edge) ÷ 2] * tan 30°.
The moment of inertia = area of equilateral triangle * ([(length of edge) ÷ 2] * tan 30°)^2
edge length 0.18 nm = side of equilateral triangle
Area = ½ * 0.18 * (0.18 * sin 60°)
(length of edge) ÷ 2] * tan 30°) = (0.18 ÷ 2) * tan 30°
The moment of inertia = ½ * 0.18 * (0.18 * sin 60°) * [(0.18 ÷ 2) * tan 30°]^2
The moment of inertia = 3.988 * 10^-5
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