Mass of a Cardioid: Find Mass & Center of Mass

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In summary, the mass of the region bounded by the cardioid r = 1 + cos(θ) is 3pi/2, and the center of mass can be found by integrating xρ and yρ with the same limits of integration, resulting in a final answer of 5pi/3. The total mass is then divided by the total mass to find the center of mass.
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Homework Statement



Find the mass and center of mass of the region bounded by the cardioid r =
1 + cos(θ), assuming the density function is given by ρ = r.

The Attempt at a Solution



The first part is simple enough, I set up the integral

[itex]\int_{0}^{2\pi}[/itex][itex]\int_{0}^{1+cos(\theta)}r dr d(\theta)[/itex]

I find the mass is 3pi/2

Now for the centre of, how do I set up the integrals? I know it would be the double integral of xρ drdθ, and then yρ drdθ, but do I keep the same limits? x would be rcos(θ) and y is rsin(θ), but this gives me some really weird functions to integrate.
 
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  • #2
I'm not sure what's so weird about the functions to integrate, but you are making a mistake already. In polar coordinates to find area you integrate the volume element dxdy=r*dr*d(theta). So I think you just computed the area. If the density is also r, that should give you an extra factor of r.
 
  • #3
Oh so in that case I get 5/6 pi. But still, what about the centre of mass? I need My and Mx, do I keep the same limits of integration?
 
  • #4
Locoism said:
Oh so in that case I get 5/6 pi. But still, what about the centre of mass? I need My and Mx, do I keep the same limits of integration?

Well, I don't get 5*pi/6. I'd check that. But, yes, keep the same limits of integration. Why not? Just put a factor of x or y into the integrand and work them out. Then you divide by total mass, right?
 
  • #5
sorry 5pi/3. Thank you!
 

1. What is a cardioid and how is it related to mass?

A cardioid is a geometric shape that resembles a heart. It is often used to describe the shape of objects or phenomena in nature, such as the path of a planet or the outline of a sound wave. In physics, the mass of an object is a measure of its quantity of matter, and it is directly related to the shape of the object. The mass of a cardioid can be determined by finding its center of mass and calculating the total mass of the object.

2. How do you find the center of mass of a cardioid?

The center of mass of a cardioid can be found by using the formula (x,y) = (a/3, 2a/3), where a is the radius of the cardioid. This formula applies to a cardioid that is centered at the origin. For a cardioid that is not centered at the origin, the center of mass can be found by breaking the shape into smaller sections and using the formula for center of mass for each section.

3. Can the mass of a cardioid change?

Yes, the mass of a cardioid can change depending on the size and shape of the object. If the radius of the cardioid changes, the mass will also change. Additionally, if the material of the object changes, the mass may also change. However, the center of mass will always remain the same as long as the shape of the cardioid remains unchanged.

4. What is the importance of finding the mass and center of mass of a cardioid?

Finding the mass and center of mass of a cardioid is important in physics and engineering fields. It allows scientists to understand the distribution of mass within an object, which can affect its stability, motion, and behavior. This information can also be useful in designing structures or machines that are balanced and efficient.

5. Are there any real-life examples of objects that have a cardioid shape?

Yes, there are many real-life examples of objects that have a cardioid shape. Some common examples include the path of a planet orbiting around a star, the shape of a sound wave produced by a tuning fork, and the shape of a heart valve. Other examples include the shape of a conch shell, the trajectory of a thrown boomerang, and the outline of a hurricane on a weather map.

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