a. Show that the first moment of a body in space about any plane through the bodys
Question:
a. Show that the first moment of a body in space about any plane through the body’s center of mass is zero.
b. To prove the Parallel Axis Theorem, place the body with its center of mass at the origin, with the line Lc.m. along the z-axis and the line L perpendicular to the xy-plane at the point (h, 0, 0). Let D be the region of space occupied by the body. Then, in the notation of the figure,
Expand the integrand in this integral and complete the proof.
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a Let the body be represented by a collection of mass points each with mass mi and position vector ri with respect to the plane The total mass of the body is M mi and the position vector of the center of mass with respect to the plane is given by R 1Mmi ri Taking the dot product of this vector with a unit normal vector n to the plane we get R n 1Mmi ri n where denotes the dot product Since ri n represents the signed distance from ri to the plane we can write mi ri n 0 because the sum of the signed distances of the mass points to the plane is zero by definition of the center of mass Therefore we have R n 0 which shows that the first moment of the body about any plane through the center of mass is zero b Let P be a point in the body with position vector r ...View the full answer
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Deepak Sharma
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Related Book For 
Thomas Calculus Early Transcendentals
ISBN: 9780321884077
13th Edition
Authors: Joel R Hass, Christopher E Heil, Maurice D Weir
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