How do we find the center of mass of a system?

Center of Mass-definition The center if mass is the average position of the mass of the system. rcm = ∑ miri / M rcm = xcmi + ycmj + zcmk

Bird toy being balanced

Baseball bat being balanced

Explain motion of dancer

Significance of finding the center of mass When an object’s center of mass accelerates, it accelerates as if all of the mass is located at the center of mass. Forces applied to points of a rigid body other than the center of mass will cause rotation.

Thought Question The figure below shows four uniform metal plates that have a section removed. The origin of the x and y axes is at the center of the original plate, and in each case the center of mass of the removed section was at the origin. In each case, where is the center of mass of the remaining section of the plate? Answer in terms of quadrants, lines, or points.

Answers to Thought Question On the y-axis between quadrant 1 and quadrant 2 In quadrant 1

Example A 4 kg mass is at the origin and a 2 kg mass is on the x-axis at x = 6.0 cm. Find the center of mass(xcm). 2 cm Xcm = 1/M(m1x1 +m2x2) Xcm = 1/6 (4(0) + 2(6) ) = 2 cm 6

Locating the Center of Mass-Problem 1 A system consists of two particles. Particle A is located at position x =1m and has a mass of 8 kg. Particle B is located at position x = 3m and has a mass of 5 kg. Where is the center of mass located? 23/13 Xcm =23/13 m 1m 3m

Locating the Center of Mass-Problem 2 Four objects are situated along the y-axis as follows: A 2 kg object is at y= +3.00m. A 3 kg object is at y= +2.50 m. A 2.5 kg object is at the origin. A 4 kg object is at y = -0.500 m. Where is the center of mass of this system of objects? Xcm = 1/M(m1y1 +m2y2 +m3y3+m4y4) 1/(2+3+2.5+4)[2(3)+3(2.5)+2.5(0)+4(0.5)]= 1m 1 m

Locating the Center of mass in 2 dimensions-Problem 3 A system consists of three particles located at the corners of the triangle as shown below. Find the center of mass of the system. x=(d +5/7 b)i y=(4/7 h)j

Locating the Center of Mass in 2 dimensions-Problem 4 A uniform piece of sheet steel is shaped. Compute the x and y coordinates of the center of mass of the piece. 11.7i+13.3j