How things balance

A very useful concept in physics is Center of Gravity (AKA CM, Center of Mass - they are usually the same point).  

Recall the demo with the weight on a stick.  Same weight, held at a further distance from the person's hand, is harder to support.  It twists your wrist more - it requires a greater "torque".

So, what is torque?

Torque - a "rotating" force

T = F L

For an object to be "in equilibrium," not only must the forces be balanced, but the torques must also be balanced.

Consider a basic see-saw, initially balanced at the fulcrum:  See image below.

If you have two people of equal weight, they should be at the same distance from the fulcrum (represented by the triangle).

You can also have two things of different weight balanced, if their distances are adjusted accordingly:  the heavier thing is closer to the fulcrum.

Mathematically, this requires that the torques be equal on both sides.

In the above problem, consider two boxes, 100 N and 300 N.  The 100 N person is 1 m from the fulcrum.  How far from the fulcrum must the 300 N box sit, to maintain equilibrium?

The calculations are worked out on the picture, but here is how it works:

Torque on left = Torque on right

100 (1) = 300 (x)

x = 1/3 m

NOTE:  The weights are NOT equal on both sides of the balance point, but the torques ARE EQUAL.  There may be a single torque on each side of a lever, or there may be multiple torques on each side.  Actually, there may be an infinite number of torques on either side.  Furthermore, there don't have to just be 2 sides - if this is a 3-dimensional (real) body, it can have an infinite number of "lever arms".  Consider a frisbee - it is round, but it still balances about a single point.  What is this special point called?  See below.

Center of Gravity

We call the "balance point" the center of gravity (or center of mass).  

• It is the point about which the object best rotates.
• It is a point where we can pretend ALL of the weight is located - for mathematical ease
• It is the point where you can apply a single force and support the object
• It is the average weighted location of mass points on the object.
• It does not HAVE to be physically on the object - think of a doughnut.
• It does NOT have to be the measured physical center of the object.
• It does NOT necessarily mean that there is equal weight on all sides of the point - think of a broom, balanced horizontally on your finger.  See below:

The person is holding the broom in a state of balance but supporting the broom at its Center of Mass.  Which side is heavier?

The right side - it is shorter, but has more mass to counter balanced the left side (longer, with left mass).  Demonstrations will be shown in class to illustrate this.

The principle is believed to originate with Archimedes (287 - 212 BC).  He is believed to have said, "Give me a place to stand on, and I will move the Earth."