Find the center of mass of a thin plate of constant density & covering the given region. The region bounded by the parabola y = 5x -5x² and the line y = -5x
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A: m1 = 1.76 kg r1 = (-1.41 , 0.619) m m2 = 5.63 kg r2 = (0.206 , -0.607) m m3 =3 kgrc.m= (-0.848 ,…
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A: position of center of mass rcm = (m1 r1 +m2 r2 )/(m1 + m2 )
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The region bounded by the parabola y = 5x-5x2 and the line y = -5x
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- Problem 3: (a) Use spherical coordinates to find the center of mass (CM) of a uniform solid hemisphere of radius R, whose flat face lies in the ry plane with its center on the origin. [Note: dV = ² sin 0 dr do do.] (b) Use your result from part (a) to calculate the CM of a hemispherical "bowl" with outer radius R and inner radius kR, k < 1. (Depending on your work in part (a), you may not even need to do another integral.) (c) Use your result from the previous part to find the CM for an infinitely thin hemispherical shell of radius R.Problem Two. A rod of length L = 2.0 m is placed on the x-axis from x = 0 to x = L. The rod has a mass density given by 2 = 2] 1+- x- where 2, = 2.0k. L' Find the location of the center of mass of the rod in meters. 4.) (A) 1.5 (B) 1.3 (С) 1.1 (D) 1.7 (E) 0.45 Consider if the rod is subjected to an applied force given by F = Aî +Bx²j , where A=10 N and B = 25 N2 . If the end of the rod at x = 0 is attached to a hinge, find the angular acceleration of the rod in rad/s². m 5.) (A) 0.68 (B) 0.92 (C) 0.26 (D) 0.34 (E) 0.45 Find the direction of the net torque vector acting on the rod. 6.) (A) +z (В) —х (C) +x (D) –z (E) -уDetermine the center of mass of a uniform cone of height h and radius R, with its axis pointing along z. (Hints: What is the CM in x and y? Let λ by the mass density per unit volume, and find the mass of the thin disk shown in the drawing. How is r/z related to R/h?)
- Find the center of mass of a rod of length L whose mass density changes from one end to the other quadratically. That is, if the rod is laid out along the x-axis with one end at the origin and the of end at x = L, the density is given by p(x) = Po + (P, - PoE where Po and e, are constant values. (Use the following as necessary: L, Po, and e,.) XCM =During the filming of a certain movie scene, the director wants a small car (whose mass is m1) traveling due east at speed v1 to collide with a small truck (whose mass is m2) traveling north. The director also wants the collision to be arranged so that just afterward, the interlocked vehicles travel straight toward the camera. If the line between the camera and collision makes an angle of theta with respect to north, at what speed v2 should the trucker drive?√9-x² and y = 25-2 Problem #7: The boundary of a lamina consists of the semicircles y together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is inversely proportional to its distance from the origin. Enter the x- and y-coordinates of the center of mass (in that order) into the answer box below, separated with a comma.