# Center of Gravity/Torque

1. Oct 11, 2004

### omin

From experiment, it's easy to see that if we balance an industrial sized broom with one finger at it's center of gravity that it remains even with the horizon. It appears parallel with the horizon.

If its sawed in half at it's center of gravity, the broom end weighs more than the broom handle end.

The reason stated in my physics book is that the broom is even with the horizon is not because of weight, but because of torque.

At first I thought the center of gravity would have to be the spot where mass was on equal sides of the spot where I balance the broom even with the horizontal axis. This would be center of mass, right?

Obviously, the midpoint of the mass of the broom is not the center of gravity. What is gravity doing internally to the mass of the broom when it creates what is called torque?

2. Oct 11, 2004

### brewnog

Right...!

When I was little, I used to play on see-saws with my little sister. Let's assume that the see-saw doesn't weigh anything, and the pivot is midway along its length.

When I weighed twice as much as my sister, she could sit twice as far away from the pivot as me, on the opposite side, and we balanced.

In a situation like this, for the see-saw to balance, the 'turning moments' (or torques) must cancel out on either side of the pivot. The turning moment is simply the product of the perpendicular force applied to the see-saw and the distance this force is from the pivot.

At the risk of patronising you, but hopefully sparing you from confusion, consider the terms "centre of mass" and "centre of gravity" to be one and the same. (They're not, but the distinction is unlikely to present itself unless you're modelling rockets or something).

Let me know if I've gone over your head/dumbed it down too much...!

3. Oct 11, 2004

### BobG

Gravity provides the force on the broom. If the force is equal on both sides of the pivot point (where you have your finger), there is no torque.

The center of mass is the point where the mass is equal on both sides of the pivot point.

Why would there be a difference? Because the force of gravity on an object depends upon that object's distance from the center of the Earth.

$$F_g=\frac{GMm}{r^2}$$

If the broom is horizontal, then the pivot point is the closest point to the center of the Earth. The ends of the broom are slightly further away from the center of the Earth and the pull of gravity on the ends is slightly less than on the pivot point. The broom is non-symmetrical. The mass on the whisk side of the broom is closer to the center of mass than the end of the broom stick, so the pull of gravity is slightly stronger even though there is the same amount of mass on either side of the pivot point.

Obviously, with an object as short as a broom, the difference between the center of mass and the center of gravity will be miniscule - much less than the width of your finger. But the difference is still there.

How big? The Earth has a radius of about 6378km. If the broom stick is about 1 meter from center of mass to the end of the broom stick, the end of the broom stick will be

$$\sqrt{6,378,000^2+1^2}$$

or about 6378km to the nearest km. In other words, you would have an extremely hard time measuring the difference experimentally for a broom stick on Earth.

Satellites do consider the difference. With so little atmosphere, little perturbations like the difference between center of mass and center of gravity start to be more significant. I can't remember the number off the top of my head, but the gravity gradient torque on the International Space Station is pretty high for environmental torques. Satellites and other spaceships (such as the shuttle) would tend to wind up with their long axis aligned with their radius if left alone (once the axis is tilted from the horizontal, the gravity gradient difference between the high side and the low side is even greater). Some of the cheaper satellites use booms on the spaceraft to create a long axis and keep their sensors pointed at the Earth (there's generally almost a fifty-fifty chance which end winds up lower, so they have to give the sensor end a tiny nudge to make sure the right end winds up pointing down).

Torque causes angular acceleration, which increases the angular velocity. In other words, the natural effect of a single boom would be for the spacecraft to swing its long axis back and forth across the radius. Damping booms that are slightly shorter than the main gravity boom keep the angular velocity slow so the satellite's long axis stays within about 5 degrees of the radius.

4. Oct 12, 2004

### omin

brewnog,

What do you mean by turning moments? I don't know what to picture.

I see the difference between centre of mass and gravity. So it's okay to consider them.

BobG,

You stated: If the broom is horizontal, then the pivot point is the closest point to the center of the Earth.

I though the pivot point was where the center of gravity is. Is that an incorrect assumption?

5. Oct 13, 2004

### SergejVictorov

I thought that the center of gravity is where the sum of all turning moments is equal on both sides. If you saw the object apart at the center of gravity, one piece could be considerably heavier than the other.

Imagine a long section of wood with a solid steel sphere at one end. The center of gravity will be found very close to the steel sphere. If you saw it apart, the part without the sphere will be nowhere as heavy as the other part. The reason why the object is balanced is because there is point mass very far from the pivot point, and since in this case:

M=m*g*R

, the torque on the left cancels out the torque on the right.

To find the center of gravity (this works only with simple objects), you can set one end of the object to the point of origin. Each point mass m1,m2,m3... will have the coordinates x1,x2,x3... and the center of gravity will be at xcg. To find xcg, apply the following formula:

xcg=(m1*x1 + m2*x2 + m3*x3.....) / (m1 + m2 + m3...)

Hope this helps a bit.

6. Oct 13, 2004

### krab

This is not obvious at all. If you suspend an object from its centre of gravity, it can have any orientation. I've done this experimentally as well. Take a long rod, see if you can balance it at a slant. Not hard to do, as long as there is sufficient friction to keep it from sliding on your finger.

7. Oct 14, 2004

### BobG

No, it is not an incorrect assumption. Both are true. The Earth is spherical and your broom is a straight line. Some point on the broom has to be closer to the Earth. If your broom is horizontal (or perpendicular to the broom's radius relative to the center of Earth), that point is your center of gravity. (This is hard to picture when you're considering normal size objects on the surface of a huge Earth. In fact, the force of gravity is considered to be uniform instead of spherical for most problems because the difference is so insignificant.)

8. Jan 4, 2007

### PhysMaster

Diagram

So does this diagram make sense according to your explanations?

Note: I have attached a .bmp file.

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