Finding the Center of Mass
In Exercises 7–-10, find the mass and center of mass of the lamina for each density.
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Calculus: Early Transcendental Functions
- Using the fact that the centroid of a triangle lies at the intersection of the triangle's medians, which is the point that lies one-third of the way from each side toward the opposite vertex, find the centroid of the triangle whose vertices are (0,0), (a,0), and (0,b). Assume a > 0 and b>0. The centroid of the triangle is (x,y), where x = and y =arrow_forwardSolve and show the proper solution. The masses mi are located in xy plane at points Pi listed below, find the center of mass.Masses: m1 = 2, m2 = 8, m3 = 5, m4 = 22Location: P1(0,0), P2(0,4), P3(5,1), P4(-1,-1)arrow_forwardCalculate the fluid force on one side of the plate The fluid force on one side of the plate is using the coordinate system shown below. Assume Ib. the density is 62.4 lb/ ft3. ... у (f) Surface of pool x (ft) -у3 — 2 Depth \y| (x,y) 7arrow_forward
- Find the center of mass of the three particles having masses of 1, 2, and 3 kg located at the points (-1,3), (2,1), and (3, -1), resp. Answer. (arrow_forwardwww Find the circulation of F = -yi + x²j + zk around the oriented boundary of the part of the paraboloid z = 9 - x² - y² lying above the xy-plane and having the normal vector pointing upward.arrow_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
- Calculate the fluid force on one side of the plate using the coordinate system shown below. Assume the density is 62.4 The fluid force on one side of the plate is lb/ft³. lb. (...) y (ft) Surface of pool ►x (ft) Depth -y = -2 |y| (x,y) - 12 Carrow_forwardDetermine the coordinates of the centroid of the shaded area in millimeters. y 2 mm у 3 (0.3 х* - 2.4 х) mm X = mm mm I| ||arrow_forwardFind a unit normal vector to the surface f(x, y, z) = 0 at the point P(-2, -5, -35) for the function -5x 5y - z f(x, y, z) = ln Please write your answer as a vector (a, b, c) with a negative z component, and show your answer accurate to 4 decimal places n= - Question Help: Video Submit Questionarrow_forward
- (c) Consider the parallelepiped with sides: u=(5,-2,1), v=(3,2, 4), and w=(-6,1,1). V (i) Find the volume of the parallelepiped.arrow_forwardhand written plz Four particles in the (x, y)-plane are held together as a rigid body by a set of light rigid rods. The particles are as described in the following table: Particle Mass (kg) Position (m) 1 1 (1, 3) 2 2 (4,2) 3 1 (3,5) 4 2 (4,1) Enter an expression that gives the moment of inertia, I, of the rigid body about the line y = x in the box below (do not include any units in your answer). I= kg m²arrow_forwardcoordinates of the centroid/center of gravity. * barrow_forward
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