Torque and Calculating the Center of Mass

Torque and the Second Condiction of Equalibrium

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This text is meant to accompany class discussions. It is not everything there is to know about static equilibrium. It is meant as a prep for class.

Click for the questions that go with this reading

Center of Gravity and Center of Mass

If a material or a body is described as being "homogeneous," then is density is the same everywhere.

Balance a ruler or meter stick on a single finger. The location where the finger is located is location of the center of gravity.

If the board broke where the finger is located then the two halves would rotate down due to the pull of gravity on each half. Gravity would create a torque on each piece.

The center of gravity is the location where the torque left of the center of gravity's location is balanced by the torque due to the right of the center of gravity's location. The symbol for center of gravity is

For a homogeneous material, the center of gravity is located at the geometric center. In the case of the ruler described earlier, the geometric center is in the middle and so is the center of gravity.

Sometimes the center of gravity is not physically on the body itself. The center of gravity for this disk toy is in the middle of the hole.

Center of Mass

The center of mass for everything done in this class is located at the same location as the center of gravity. The only time it is not at the same location is when the gravitational field varies greatly across the size of the object. The Moon is an example of this.

The center of gravity and center of mass begin to move apart when you look at large objects in space. For example, suppose the Moon is homogeneous, in reality it's not, and a perfect sphere. The geometrical center would be at the center of the Moon. Therefore the center of mass is located at the Moon's center. The center of gravity is not at the center. The moon is so large that the Earth's pull on the side facing the planet is greater than the pull on the side opposite the Earth. This will move the center of gravity away from the Moon's center and closer to the Earth.

Here is my YouTube video describing how the center of mass is found using torques. It contains the formula needed to find the center of mass and center of gravity.

This video can be found on YouTube here, http://www.youtube.com/watch?v=cJy3rsqvqDg

Practice

Below is a simple problem showing how the center of gravity and center of mass is calculated.

Question
Paper Solution
Video Solution

A uniform beam of 5.0 kg holds up the two masses as shown. Find the center of gravity of the system of objects from the left edge of the board.

Image of the setup

This video can be found on YouTube at http://youtu.be/Uhdlx668tNI

Question
Hint
Solution

The Earth's mass is 5.972 x 10²⁴ kg. The Moon's mass is 7.0349 x 10²²kg. The average distance between the center of the Earth and center of the Moon is 3.78 x 10⁸ m. How far from the center of the Earth is the location of the center of mass of the two bodies?

by Tony Wayne ...(If you are a teacher, please feel free to use these resources in your teaching.)

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