# Finding Center of Mass of Disc with Hole

• Aliasa
In summary: So in the end, I had to look it up. In summary, the center of mass of a metal disc with a hole is located at (-1/30,0).
Aliasa

## Homework Statement

Find the center of mass of a uniform sheet in the form of a circular disc
with a hole bounded by the equations x^2 + y^2 ≤ 1 and (x - 1/2)^2 + y^2 ≥ (1/16).

## The Attempt at a Solution

Can you find the area of the figure?

That is all the info given.

Aliasa said:
That is all the info given.

Hi Aliasa. http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif

Using your knowledge of mathematics, sketch the shape. Then post it here.

Last edited by a moderator:
SteamKing said:
Can you find the area of the figure?

Aliasa said:
That is all the info given.
Yes, but can you find the area? This is a disk with a hole in it. It would be a good idea to at least draw a picture of the "hole bounded by the equations x^2 + y^2 ≤ 1 and (x - 1/2)^2 + y^2 ≥ (1/16)." Those boundaries are circles. Can you graph the circles?

Where is the center of mass of the outer circle located?
Where is the "center of mass" of the hole located?
What is the area of the outer circle?
What is the area of the hole?

It turns out the question is horribly worded. The sheet is bounded by the former equation, while hole by the latter -_-. Since the equations are inequality, by taking density into account, I feel the center of mass should be a range too. The hole can have a max radius of .5, sheet, 1. Hole can have a min radius of 1/4, when the sheet can have a minimum of 3/4.

Aliasa said:
It turns out the question is horribly worded. The sheet is bounded by the former equation, while hole by the latter -_-. Since the equations are inequality, by taking density into account, I feel the center of mass should be a range too. The hole can have a max radius of .5, sheet, 1. Hole can have a min radius of 1/4, when the sheet can have a minimum of 3/4.
You are misunderstanding the question. The metal template is very precisely defined by the two equations; there is no range of possible shapes and sizes. The centre of mass is an exact point. The two equations together define the metal surface, the hole is where there is metal missing.

Have you sketched the shape yet?

If that is the case then the question is trivial. I have solved it if those equations represent what you say. But doing that only takes me 5 minutes or less, which I can't understand. All the other questions on the assignment take in excess of 3 hours.

Aliasa said:
If that is the case then the question is trivial. I have solved it if those equations represent what you say. But doing that only takes me 5 minutes or less, which I can't understand. All the other questions on the assignment take in excess of 3 hours.

That's pretty astounding. Can you provide an example of one of these 3-hour problems?

Aliasa said:
Yes. That's correct. Nice job.

Chet

This is one of those questions. Others seem easy now that I have done them. Yet to start on this one. The astounding thing is there's no mention of 'coefficient of restitution' in lecture notes.

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## 1. What is the center of mass of a disc with a hole?

The center of mass of a disc with a hole is the point at which the disc would balance if it were placed on a pivot at that point. It is the average position of all the mass in the disc.

## 2. How is the center of mass of a disc with a hole calculated?

The center of mass of a disc with a hole can be calculated using the formula: x = (M1x1 + M2x2) / (M1 + M2), where x is the position of the center of mass, M1 and M2 are the masses of the disc and the hole respectively, and x1 and x2 are the distances of the disc and the hole from a reference point.

## 3. Does the shape and size of the hole affect the center of mass of a disc?

Yes, the shape and size of the hole do affect the center of mass of a disc. The closer the hole is to the edge of the disc, the more it will shift the center of mass towards it. Similarly, a larger hole will have a greater effect on the center of mass compared to a smaller hole.

## 4. Can the center of mass of a disc with a hole lie outside the disc?

No, the center of mass of a disc with a hole will always lie within the boundaries of the disc. This is because the center of mass is calculated by taking into account the positions and masses of all the particles within the disc, and the hole is a part of the disc's boundary.

## 5. How is the center of mass of a disc with a hole used in practical applications?

The concept of the center of mass of a disc with a hole is used in various practical applications, such as in designing objects with balanced weight distribution, determining the stability of structures, and calculating the motion of objects. It is also useful in physics and engineering calculations involving rotational motion and torque.

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