# Center of Mass

by Nickolas C. Jensen June 1998

Angular Mechanics I

• Here are the quantities you will soon be able to calculate:
• mass M
• distance X

These quantities can be in any unit (i.e. lbs, kg, g, for mass, and cm, m, in, ft for distance)

• Defining Center of Mass

The Center of Mass is essentially the balance point. In order to find the Center of Mass, we must find the average of the masses and the average distance, as we are trying to find the middle, or center of mass. The Center of Mass can also be thought of as the Center of Gravity, or the point where gravity acts on an object. Of course gravity acts on all parts of the object, but it is said to be focused at the Center of Mass. The simplest example is that of a teeter totter. To find the Center of Mass when two people sit on a teeter totter, you multiply the mass of each individual times the distance that they are sitting from the center of the teeter totter, giving you the equation M1X1 = M2X2. In a slightly more complex situation, if you have several masses balancing on a rod, you multiply the masses by their distance from a fixed point, add the quantities together, and then divide by the sum of the masses, using the equation:

Center of Mass (COM) = (M1X1 + M2X2 + M3X3 + . . . MnXn)/(M1 + M2 . . . Mn)

It's all really easy, try some example problems and you'll see.

Here is a list of the all of the formulas we will be using:

• M1X1 = M2X2 (Teeter Totter)
• COM = (M1X1 + M2X2 + . . .MnXn)/(M1 + M2 + . . . Mn) (Center of Mass)

1. Go through the problem and figure out what is given or implied
Make a list, and identify the quantities you know.
2. Find any formula that will allow you to calculate
anything that you don't know, and apply it.
3. Add what you just found in the last step to your list of knowns.
4. Check to see if you have found the answer. If not, repeat the
previous two steps until you are done.

## Example problem 1

1. If a 120 kg Gramps sits 8.5 feet from the pivot of a see-saw, how far should 30 kg Junior sit to balance?
1. We will use the equation M1X1 = M2X2, It is a basic ratio, with (120)(8.5) = (30)(X2). We then divide 30 into 1020 to get X2. The answer is 34 feet, so Junior should sit 34 feet from the center of the teeter totter to balance with Gramps.

## Example problem 2

1. How far is the Center of Mass from the larger of a 12 lb bowling ball and a 10 lb bowling ball that are 50 cm distant?

We will use the second equation for this one. The only tricky part to this problem is knowing where the fixed point is. In this problem the fixed point is at the 12-lb bowling ball, so the equation is as follows: ((12)(0) + (10)(50))/(12 + 10) = COM. Note that the 12 lb ball is at distance 0 cm. The answer is about 22.7 cm.

## Sample Problems

link to a solution to the problem. Try the problem, check your
answer, and go to the solution if you do not understand.

## 1.

How far is the Center of Mass from a 34 kg object if it is 13 cm from a 78 kg object?

## 2.

A 69 lb child sits 5 ft from the pivot point of a teeter totter, how far should their 130 lb father sit to balance on the other side?

## 3.

Kyle and Debbie balance on a teeter totter. Debbie weighs 110-lbs and is sitting 8 ft from the pivot point, and Kyle is sitting 6.3 ft out. What is his mass?

## 4.

The COM between two objects is 12 cm from the one with a mass of 34 kg. What is the mass of the other one if it is 56 cm from the COM?

(7.3 kg)

## 5.

A 5 kg mass is on the 0 end of a meter stick, and a 3 kg mass is on the 100 end of the stick. Where is the COM? Neglect the mass of the meter stick.

## 6.

A 165 lb and a 120 lb person sit on a see-saw that is 32 ft long. How far is the balance point from the lighter person?

## 7.

How far is the COM of the sun and Jupiter from the center of the sun? The mass of the sun is 1.99E30 kg and the mass of Jupiter is 5.98E24 kg. Look up the distance between the sun and Jupiter, and the radius of the sun.

## 8.

Devise a way to find the COM of any triangle using a straight edge, a compass, and a divider. Explain it.

## 1.

For this question, it is actually easier to think of it in terms of the second equation. Let's put the numbers that we know into the equation.

• ((34 kg)(0 cm)+(78 kg)(13 cm))/(34 kg+78 kg)
• In this equation, we have all of the variables for our formula, so you just need to calculate them. The answer is 9.05 cm.

## 2.

Here's what we know:

• M1 = 69 lbs
• X1 = 5 ft
• M2 = 130 lbs
• This question is easy to solve with the first equation.
• We plug in the values to get (69 lbs)(5 ft) = (130 lbs)(X2)
• so (345 ft-lbs) = (130 lbs)(X2)
• divide 345 by 130 to get approx. 2.65 ft.

## 3.

• M1 = 110 lbs
• X1 = 8 ft
• X2 = 6.3 ft
• Again, we use the first equation M1X1 = M2X2
• 110(8) = 6.3(M2)
• We divide 6.3 into 880 to get about 139.68, or 140 lbs

## 5.

• Using the second equation, we get:
• ((5)(0) + (3)(100))/(5+3) = COM
• Remember that the mass of the meter stick is negligible.
• The answer is 37.5 cm

• 6.
• The fixed point is at the lighter person, because we are trying to find where the COM is from them.
• Therefore, using the second equation, we can find:
• ((120)(0) + (165)(32))/(120+165) = COM
• The answer is 18.5 ft

• 7.
• A .12 m radius disk sander has a motor that spins the disc at 1200 rpm. a) What is the angular velocity in rad/sec? b) What is the tangential velocity of the disk at the edge of it?