# Centroid & Center of Mass of a Semicircle Video

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• 0:04 Definition of a Centroid
• 0:51 Finding Centroid of a…
• 1:40 Some Practice Problems
• 2:39 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

The centroid is the point at the exact center of an object. If the object has a uniform density, then the center of mass will be located at the centroid. In this lesson, learn how to find the centroid of a semicircle.

## Definition of a Centroid

The point that is in the exact center of a shape or object is known as the centroid. If the object has a uniform density, then the centroid is also the center of mass, which is the point at which the object will balance with just as much mass on all sides.

Just like the pencil, you also have a center of mass, and you have to be careful to keep it directly over your feet or you'll lose your balance too! To see how this works, stand up with your back against a wall and try to lean forward until you touch the floor. You will find that you can't do it without falling over! This happens because as you lean forward, your center of mass moves further and further in front of your body until it's not over your feet anymore, and you lose your balance, just like the pencil does if you don't balance it exactly at the center of mass.

## Finding Centroid of a Semicircle

All shapes have centroids. For very simple shapes like rectangles or circles, it's easy to find the centroid because it's just in the middle of the shape. However, for more complex shapes, like semicircles, you may not know exactly where the center is.

To find the centroid of a semicircular shape, you need to know the radius (r), and then the x- and y-coordinates of the centroid can be found as shown on screen now:

Did you notice that the x-coordinate of the centroid is zero? That is because the coordinate system is placed in the center of the semicircle.

It's certainly possible to find the centroid using other coordinate systems. The most common is probably one placed at the edge of the semicircle. How would the location of the centroid change in that case?

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