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Module MA1132 (Frolov), Advanced Calculus Tutorial Sheet 9 To be solved during the tutorial session Thursday/Friday, 24/25 March 2016 You may use Mathematica to sketch the integration regions and solids, and to check the results of integration. Moment of inertia: The tendency of a solid to resist a change in rotational motion about an axis is measured by its moment of inertia about that axis. If the solid occupies a region G in an xyz-coordinate system, and if its density function δ(x, y, z) is continuous on G, then the moments of inertia about the x-axis, the y-axis, and the z-axis are denoted by Ix , Iy , and Iz , respectively, and are defined by ZZZ (y 2 + z 2 )δ(x, y, z)dV , Ix = Z Z ZG (x2 + z 2 )δ(x, y, z)dV , Iy = (1) G ZZZ Iz = (x2 + y 2 )δ(x, y, z)dV . G Newton’s law of gravitation: Let a solid occupy a region G in an xyz-coordinate system, and let its density function δ(x, y, z) be continuous on G. Then the gravitational force F = Fx i + Fy j + Fz k exerted by the solid on a point particle of mass m located at (ξ, η, ζ) is given by ZZZ x−ξ Fx (ξ, η, ζ) = Gm δ(x, y, z)dV , r3 Z Z ZG y−η δ(x, y, z)dV , Fy (ξ, η, ζ) = Gm r3 (2) Z Z ZG z−ζ Fz (ξ, η, ζ) = Gm δ(x, y, z)dV , 3 G r p r = (x − ξ)2 + (y − η)2 + (z − ζ)2 , where G is the gravitational constant. The force can be obtained from the gravitational potential field U (ξ, η, ζ) as follows ZZZ 1 U (ξ, η, ζ) = −G δ(x, y, z)dV , G r ∂U (ξ, η, ζ) ∂U (ξ, η, ζ) ∂U (ξ, η, ζ) Fx (ξ, η, ζ) = −m , Fy (ξ, η, ζ) = −m , Fz (ξ, η, ζ) = −m . ∂ξ ∂η ∂ζ (3) In what follows we set m = 1, G = 1, and consider homogeneous solids with δ(x, y, z) = 1. 1. Consider the solid G bounded by the surfaces z = 0, z = h and and x2 + y 2 = a2 . (a) What are the surfaces z = 0 and z = h? 1 (b) What is the surface x2 + y 2 = a2 ? (c) Sketch the solid G. (d) Sketch the projection of the solid G onto the xy-plane. (e) Find the volume V of the solid G. (f) Find the centroid of the solid G. (g) Find the moments of inertia of the solid G. (h) Find the gravitational force exerted on a point particle by the solid G if the point particle is located at the origin (0, 0, 0). Find its limit as a → ∞ with h kept fixed, and its limit as h → ∞ with a kept fixed. Show the details of your work. 2. Consider the solid G bounded above by the surface 1 15 p 2 − x + y 2 and below by the surface z = (x2 + y 2 ) . 2 2 p (a) What is the surface z = 15 − x2 + y 2 ? 2 z= (b) What is the surface z = 21 (x2 + y 2 )? (c) Sketch the solid G. (d) Sketch the projection of the solid G onto the xy-plane. (e) Find the volume V of the solid G. (f) Find the mass M of the solid G if its density is √ 2 2 e x +y − 1 p . δ(x, y, z) = x2 + y 2 + 5 x2 + y 2 (g) What is the density of the solid G at the origin: δ(0, 0, 0) =? Show the details of your work. 2