# Find the location of the CM of a hollow ice cream cone

• Tonyt88
In summary, the location of the center of mass for a hollow ice cream cone with base radius R and height h and uniform mass density can be found using the formula [h^2]/[(3)(R+sqrt(R^2 + H^2))]. If the cone is solid, the formula changes to h/4. The angle that the cone makes with its symmetry axis is given by \tan\theta=R/h.
Tonyt88
Find the location of the CM of a hollow ice cream cone, with base radius R and height h, and uniform mass denisty. How does your answer change if the cone is solid, instead of hollow?

Okay, so I'm pretty sure that I need to work with slices, and that you need the mass which I believe is [where sigma is density]

σ ( (pi) r^2 + (pi) r √r2 + h2)

Though I don't know where to go from here

Nm, I think I got it, I have:

[h^2]/[(3)(R+sqrt(R^2 + H^2))]

and for the filled cone:

h/4

$$\vec{R}_{CM}=\frac{\int_A\sigma\vec{r}dA}{\int_A\sigma dA}$$

Notice that the angle that the cone makes with its symetry axis is $\tan\theta=R/h$. I leave it to you to evaluate the integrals.

Tonyt88 said:
Nm, I think I got it, I have:

[h^2]/[(3)(R+sqrt(R^2 + H^2))]

and for the filled cone:

h/4
The first one looks far too complicated. What is H anyway? Is that where the CM is located? The filled cone looks good.

## 1. How do you find the location of the CM of a hollow ice cream cone?

To find the location of the center of mass (CM) of a hollow ice cream cone, you need to consider the cone as a combination of two objects: the cone itself and the hollow space inside. The CM of the cone can be found using the formula for the CM of a cone, while the CM of the hollow space can be found using the formula for the CM of a hollow cylinder. The overall CM can then be found by taking the weighted average of the two CMs.

## 2. Why is it important to find the location of the CM of a hollow ice cream cone?

Knowing the location of the CM of a hollow ice cream cone is important for various engineering and physics applications. It can help in designing stable structures, understanding the motion of the cone when it is thrown or dropped, and determining the amount of force needed to balance the cone at a certain point.

## 3. What factors affect the location of the CM of a hollow ice cream cone?

The location of the CM of a hollow ice cream cone is affected by the size and shape of the cone, as well as the distribution of mass within the cone. It is also influenced by the density of the ice cream and any toppings or fillings inside the cone.

## 4. Can the location of the CM of a hollow ice cream cone change?

Yes, the location of the CM of a hollow ice cream cone can change depending on the position of the cone. For example, if the cone is tilted or held at different angles, the CM will shift accordingly. The addition or removal of materials inside the cone can also affect the location of the CM.

## 5. How does the location of the CM of a hollow ice cream cone affect its stability?

The location of the CM of a hollow ice cream cone is directly related to its stability. If the CM is located above the base of the cone, it will be stable and remain upright. However, if the CM is located outside the base, the cone will be unstable and will topple over. This is why it is important to find the location of the CM when designing structures or balancing objects.

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