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Think About It The center of mass of a solid of constant density is shown in the figure. In Exercises 43-46, make a conjecture about how the center of mass
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- Which lines or line segments or rays must be drawn or constructed in a triangle to locate its a orthocenter? b centroid?arrow_forwardAstronomers use a technique called stellar stereography todetermine the density of stars in a star cluster from theobserved (two-dimensional) density that can be analyzedfrom a photograph. Suppose that in a spherical cluster ofradius R the density of stars depends only on the distancefrom the center of the cluster. If the perceived star density isgiven by y(s) , where is the observed planar distance fromthe center of the cluster, and x (r ) is the actual density, it canbe shown that y(s) = integral s to r 2 r /sqrt ( r2 - s2 ) x(r) dr If the actual density of stars in a cluster is x (r ) = 1/2 (R-r)2 ,find the perceived density y(s)arrow_forwardHow do you find the center of mass?arrow_forward
- 人工知能を使用せず、 すべてを段階的にデジタル形式で解決してください。 ありがとう SOLVE STEP BY STEP IN DIGITAL FORMAT DON'T USE CHATGPT 12. Find the moments of inertia, as well as the radii of gyration in x and y for the triangle 0 ≤ x < b, 0≦y shassuming that the density is constant, o = 0_0. 13. Find the centroid and the moments of inertia Ix, ly and Iz of the tetrahedron whose vertices are the points (0, 0, 0), (1, 0, 0), (0, 1, 0) and (0, 0, 1).arrow_forwardDetermine the type of points on the X (u, v) = (u, v, u?) surface. Differential geometryarrow_forwardPlease provide Handwritten answer. Advanced Math We consider a thin plate occupying the region D located in the upper half-plane (where y ≥ 0) and between the parabolas of equations : y = 2 - x2 and y = 1 - 2x2 The density of the plate is proportional to the distance from the x axis. a) Calculate the moments of inertia (second moments) of the plate with respect to the coordinate axes.b) Is it easier to rotate the plate around the x-axis or the y-axis? Justify your answer.arrow_forward
- Q1,- A- Locate the centroid (X only) of the shaded area y=12 -3x 3 m Fig. L.A 2 marrow_forwarddivĒ = 0 divĤ = 0 1 ƏĦ curlĒ c ðt 1 DỂ curlĦ - c ət Using Maxwell's equations, show 1 ²Ē c2 Ət² (V · V)Ẽ =arrow_forwardThe vertical stern of a boat partially submerged in seawater with a superimposed coordinate system is shown in the figure. The table shows the widths w of the stern (in feet) at indicated values of y. Use the Midpoint Rule with n = 4 to approximate the fluid force against the stern.arrow_forward
- y = ex, y = 0, x = 0, x = 3 Find the exact coordinates of the centroid. (x, y) =arrow_forwardA point charge, Q=2 µC is located at A (4, 3, 5) in free space. Calculate E,, Es and E, at the point P (8, 12, 2). (a)arrow_forwardSeawater has density 1025 kg/m³ and flows in a velocity field v = yi + x j, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the hemisphere x2 + y2 + z2 = 64, z 2 0. kg/sarrow_forward
- Elementary Geometry For College Students, 7eGeometryISBN:9781337614085Author:Alexander, Daniel C.; Koeberlein, Geralyn M.Publisher:Cengage,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage