Finding the Center of Mass (or gravity)

Simplifying a system When we have a system such as a group of objects, or a complex structure with different parts (like a hammer, or broom) we often find it easier to simply that system into a single object whose mass is the total mass of the system. But this mass of this “new object” must have a location, this is the average location of all the individual locations of each mass that made up the system. Just like your average grade is the average of all your individual grades, and just like for your grades this average location is a weighted average. We call this average location of the system’s mass the center of mass for the system.

Calculating center of mass The equation for center of mass is…. (Total mass of system)*(Center of mass) = The sum of each mass times it’s location (  M)(X) =  (M i X i ) All locations for X must be measured from the same reference point If the location of the mass is to the right of the refrecne the value is + + often means to the right or up If the location of the mass is to the left of the reference the value is – - often means to the left or down

Sample Problem Find the center of mass of the system as seen by the 5 kg mass. 10 kg 5kg15 kg 4 m 2 m

Step 1: Set Locations for each mass 10 kg 5kg15 kg 4 m 2 m X 5kg = 0m (this is our reference) X 10kg = -4m (this is 4m to the left of our reference) X 15kg = +2m (this is 2m to the right of our reference)

Step 2: Use the equation 10 kg 5kg15 kg X 5kg = 0m X 10kg = -4mX 15kg = +2m (30 kg)(X) = (10 kg)(-4m) + (5kg)(0m) + (15kg)(+2m) (30 kg)(X) = (-40 kg*m) + (0kg*m) + (+30kg*m) (30 kg)(X) = (-10 kg*m) (X) = (-10 kg*m)/(30 kg) (X) = (-.333 m) The center of mass of the system is.333 meters to the left of the 5 kg mass.

For rods and large objects Although the force of gravity truly acts many small forces pulling on all parts of an object, we often sum all the forces and considered there to be one gravitational force acting on one point called the center of gravity (AKA center of mass) Centers of mass for uniform objects are located in the center of the object. FgFg Center of mass

Sample Problem Find the center of mass of the system as seen by point P. 4 m 10 kg P 5 kg

Step 1: 4 m 10 kg P 5 kg X p = 0m (this is our reference) X 5kg = +4m X 10kg = +2m (the rods center of mass is in it’s middle so we can simply the rod into a single point, or object, whose mass location is 2m to the right of pint P) 2 m

Step 1: 4 m P 5 kg X p = 0m (this is our reference) X 5kg = +4m X 10kg = +2m (the rods center of mass is in it’s middle so we can simply the rod into a single point, or object, whose mass location is 2m to the right of pint P) 2 m 10 kg

Step 1: P 5 kg X p = 0m X 5kg = +4mX 10kg = +2m 10 kg (15 kg)(X) = (10 kg)(+2m) + (5kg)(+4m) (15 kg)(X) = (+20 kg*m) + (+20 kg*m) (X) = (+40 kg*m)/(15 kg) (X) = (+2.67 m) The system’s center of mass is 2.67 meters To the right of point P

Center of mass and balance point The center of mass of a system is it’s balance point. So if you known where the balance point is you already know where the center mass is. And if you are looking for the balance point, you are really looking for the center of mass.

Center of mass for the Y axis Just like for the X axis a system has a center of a mass for the Y axis. We find the Y axis center of a mass the same way as we found the center of mass for the X axis (Se simple use Y locations instead of the X locations) Locations above the reference are + values Locations below the reference are - values The equation for center of mass is…. (Total mass of system)*(Center of mass) = The sum of each mass times it’s location (  M)(Y) =  (M i Y i )