What is the Pappus Theorem and how can it be used in physics?

[WiFO215]

Lets start a thread for all the math knick-knacks you learned to do physics. I'll start. Reading the Feynman Lectures,
For finding the centre of mass.
Using the pappus theorem when a 2D object is rotated about a plane that is perpendicular to the ones it exists in i.e. mutually perpendicular to the planes an object exists in, and it is rotated in this plane, the distance traveled by the centre of mass times the area of the 2D object is the volume of the 3D object generated by rotation.

For example,
taking a semicircle of radius r, let the COM be a distance X from the flat side of the semicircle. When you rotate about the flat side,
distance traveled by COM = 2 x pi x X
Area of semicircle = pi x r^2 x 1/2
Area of 3D object generated = 4/3 x pi x r^3

When you equate, you get X = 4r/3pi

I thought this was fantastic!

 

[AndyW]

Thanks for starting this thread! I learned a similar concept in my physics class, but with a different application. We used the concept of moment of inertia to find the rotational kinetic energy of an object. The moment of inertia is the resistance of an object to rotational motion and is calculated by summing the products of mass and squared distance from the axis of rotation.

For example, in a uniform solid sphere rotating about its center, the moment of inertia is given by I = (2/5)MR^2, where M is the mass of the sphere and R is the radius. This concept can also be extended to more complex objects by using the parallel axis theorem, which states that the moment of inertia about an axis parallel to the original axis of rotation is equal to the moment of inertia about the original axis plus the product of the mass and squared distance between the two axes.

I found this concept really useful in understanding the rotational motion of objects and how it relates to their mass and distribution of mass. It's amazing how math can help us understand and describe the physical world around us!

 

[sir-pinski]

The Pappus Theorem is a mathematical theorem that relates the volume of a three-dimensional object to the area of its cross-section and the distance traveled by its center of mass during rotation. It states that the volume of a three-dimensional object generated by rotating a two-dimensional object around a fixed axis is equal to the product of the area of the two-dimensional object and the distance traveled by its center of mass.

In physics, this theorem can be used in various situations. One application is in finding the moment of inertia of an object. The moment of inertia is a measure of an object's resistance to rotational motion, and it is an important concept in physics. The Pappus Theorem can be used to calculate the moment of inertia of an object by rotating it around a fixed axis and using the theorem to relate the volume of the 3D object generated to its moment of inertia.

Another application of the Pappus Theorem in physics is in finding the center of mass of an object. The center of mass is the point at which an object's mass is concentrated, and it plays a crucial role in understanding an object's motion. By using the theorem, we can relate the distance traveled by the center of mass during rotation to the object's volume and use this information to calculate the center of mass.

Overall, the Pappus Theorem is a useful tool in physics that allows us to relate the geometry of an object to its physical properties, such as its moment of inertia and center of mass. It is just one of the many mathematical concepts that have applications in the world of physics and helps us to better understand the physical world around us.