Center of Mass Center of Mass The center

Center of Mass

Center of Mass The center of mass of a system is the point at which all the mass can be assumed to reside. Sometimes the system is an assortment of particles and sometimes it is a solid object. Mathematically, you can think of the center of mass as a “weighted average”.

Center of Mass – system of points

Practice Problem: Find the Center of mass of the system as (x, y) y 2 Center of mass = (4, -1) 2 kg x 0 3 kg 1 kg -2 -2 0 2 4

Center of Mass of solid objects • If the object is of uniform density, you pick the geometric center of the object. x x x • For combination objects, pick the center of each part and then treat each center as a point. x x

Example: An “L” Shaped Object a 2 a One approach is to divide the object up into two separate objects: 2 a I a a II a Mass = ρV = ρat (t = thickness) m. I = ρ (2 a x a) t = 2ρa 2 t m. II = ρ (a x a) t = ρa 2 t Rcm = 5/6 aî +5/6 aĵ

Example: An “L” Shaped Object a 2 a Another approach is treat an area as if it had negative mass: 2 a I 2 a a II a Mass = ρV = ρat (t = thickness) m. I = ρ (2 a x 2 a) t = 4ρa 2 t m. II = ρ (a x a) t = ρa 2 t Rcm = 5/6 aî +5/6 aĵ

Practice Problem: Disk with hole y R Determine the center of mass of this shape. 2 R x If you took the origin to be in the far left corner: Rcm = 7/3 Rî +7/3 Rĵ If you took the origin the be in the center of the large circle: Rcm = 1/3 Rî +0ĵ

Center of Mass for more complicated situations • If the shapes are points or simple geometric shapes of constant density, then • Anything else is more complicated, and

Example Problem: Rod with constant linear mass density y Find the x-coordinate of the center of mass of a rod of length L whose mass per unit length = l (where l = M/L) If you divide the rod into equal segments of length dx, the mass of each segment would be: dm = λdx Because l = M/L, this reduces to: L x

Practice Problem: Rod with constant linear mass density Find the x-coordinate of the center of mass of a rod of length L whose mass per unit varies according to the expression l = ax (where a is a constant) y L x

Motion of a System of Particles If you have a problem involving a system of many particles, you can often simplify your problem greatly by just considering the motion of the center of mass. If all forces in the sytem are internal, the motion of the center of mass will not change. The effect of an external force that acts on all particles can be simplified by considering that it acts on the center of mass; that is: SFext = Macm