Question 9: A half uniform annulus of inner and outer radii R1 and R2 has a non- uniform surface mass density given by o = 0or where oo is a positive constant and r is the radial distance from the origin (radial coordinate in the cylindrical coordinates) as shown in Figure 6. Find i) the mass of the half annulus, and ii) the position of the center of mass, i.e., (xem, Yem) the center of mass in terms of R1 and R2.
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- Four beads, each of mass m = 1 kg, are attached at various 1 kg, and radius R= 1 m (see figure). Find the coordinates of the center of locations to a ring, also of mass m M mass of the system consisting of the ring and the beads. Angle A, located between the first bead and the horizontal, is equal to 47°. Angle B, located between the horizontal and т, the second bead, is equal to 50°. Angle C, located between the third bead and the horizontal, is equal to 68°. Angle D, D located between the horizontal and the fourth bead, is equal х to 31°. Хст т, Ycm = т,Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. 7 y = 1 + x2 y = 0 X = -1 X = 1 p = k m = (x, y) = Need Help? Watch It Read ItTwo objects, A and B, collide. Consider the magnitude of the force that object A exerts on object B, and vice versa. Predict whether you expect the two forces to be the same or different, and explian why.
- A 3 kg mass is located at (-3, 3), a 7 kg mass is located at (4, 2) and a 1.2 kg mass is located at (1, -1). (a) Calculate the x coordinate and y coordinate of the center of mass for the mass system. (b) What happens to the center of mass as the 1.2 kg mass is gradually increased?Please do part bA uniform soda can of mass 0.141 kg is 12.1 cm tall and filled with 0.351 kg of soda (figure below). Then small holes are drilled in the top and bottom (with negligible loss of metal) to drain the soda. (Initially the soda can is full.) (a) What is the height h of the com of the can and contents initially? cm (b) What is the height h of the com of the can and contents after the can loses all the soda? cm (c) What happens to h as the soda drains out? O decreases then rises again decreases to the bottom O rises to the top O stays the same (d) If x is the height of the remaining soda at any give instant, find x when the com reaches its lowest point. cm Splash!
- 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) -уTwo thin frames are fixed with light bars with a mass, so that the center of the two frames applies. The moment of failure of the system around an axis that passes from the center and perpendicular to the page level is equal to: Choose and solve step
- 11.) see photo for details, please provide full body diagram with solutionNeeds Complete typed solution with 100 % accuracy.Determine 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?)