Composite Bodies: Finding the Centroid

In summary, the problem involves finding the z-bar centroid for a composite body made of a single scoop of ice cream placed into a cone. The scoop of ice cream is a sphere with a radius of 3.65 cm, while the cone has an exterior radius of 3.10 cm and an interior radius of 2.80 cm. The cone is hollow inside and has a height of 9.90 cm, with an interior height of 9.00 cm. After finding the product of volume and centroid for both the sphere and the cone, the correct method is to subtract the product of volume and centroid for the inner cone (considered solid) from the product of volume and centroid for the entire cone. The z
  • #1
Cottontails
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Homework Statement


1119614_002.jpg

A single-scoop ice cream cone is a composite body made from a single scoop of ice cream placed into a cone. Assume that the scoop of ice cream is a sphere with radius r = 3.65 cm that is placed into a 9.90 cm tall cone. The interior height of the cone is 9.00 cm. The cone has an exterior radius of 3.10 cm and an interior radius of 2.80 cm. The scoop of ice cream sits on the cone's interior radius and extends into the cone some distance. Find the z-bar centroid for the cone (the scoop of ice cream and the cone).

Homework Equations


Components: Sphere (scoop of ice-cream) and Cone (ice-cream cone)

Volume of Sphere: V=4/3∏R3, V=4/3∏(3.65)3, V=203.6888249
Centroid of Sphere: 12.2
Product of Volume and Centroid: 12.2 x 203.6888249 = 2485.003664

Volume of Cone: V=1/3∏R2h, V=1/3∏(3.10)2(9.90), V=99.62932782
Centroid of Cone: 6.779
Product of Volume and Centroid: 99.62932782 x 6.779 = 675.3872133

The Attempt at a Solution


So, z-bar = 10.40 cm (Sum of product of volume and centroid for sphere and cone/Sum of volume of sphere and cone)

I tried that answer however, it was wrong.
I then realized that the cone is not a completely solid shape, with it being hollow inside and that it has two different radii and height.
As this image shows:
1119614_003.jpg

So, I then found the product of volume and centroid for the inner cone: volume being 73.89025921 and centroid being 6. Then, I subtracted this from the other product of volume and centroid for the cone and then found z-bar again. This was also wrong.

I'm not sure with what else there is to do. Any help, please? I've used all the hints for the question and I only have one attempt remaining.
 
Last edited:
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  • #2
Cottontails said:
View attachment 155785
So, I then found the product of volume and centroid for the inner cone: volume being 73.89025921 and centroid being 6. Then, I subtracted this from the other product of volume and centroid for the cone and then found z-bar again.
This is the right idea. But, the centroid for the inner cone (considered solid) is not located at z = 6. It is located at z = 0.90 cm +(3/4)(9.00 cm) = 7.65 cm
 

1. What is the centroid of a composite body?

The centroid of a composite body is the geometric center of the body, where all the mass is evenly distributed. It is often referred to as the "center of gravity" or "center of mass".

2. How is the centroid of a composite body calculated?

The centroid of a composite body can be calculated by dividing the sum of the individual moments of each component by the total mass of the body. This can be done using the formula: x̄ = ∑(mi * xi) / ∑mi and ȳ = ∑(mi * yi) / ∑mi, where mi is the mass of each component and xi and yi are the respective coordinates of the component's centroid.

3. What is the significance of finding the centroid of a composite body?

Finding the centroid of a composite body is important in engineering and physics as it helps determine the stability and balance of the body. It also helps in the design and analysis of structures, as the centroid is where the body will rotate or balance.

4. Can the centroid of a composite body be outside of the body?

Yes, the centroid of a composite body can be outside of the body. This can happen when the mass distribution of the body is uneven or when there are empty spaces within the body. In these cases, the centroid will still represent the center of mass, but it may not be located within the physical boundaries of the body.

5. How can knowing the centroid of a composite body help in real-world applications?

Knowing the centroid of a composite body can be helpful in various real-world applications, such as in the design of buildings, bridges, and other structures. It can also be useful in determining the stability and maneuverability of vehicles and aircraft. Additionally, it is important in understanding the behavior of objects under different forces, such as in sports equipment or machinery.

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