Center of mass of semicirculardisk calculated from center of mass of semicircular arc

In summary: C.M of the semi circular disk. So you can use the centroid formula for triangles to calculate the C.M of the arc. Then use this value to calculate the C.M of the semi circular disk.In summary, in order to calculate the center of mass (C.M.) of a semi-circular disk with radius r and mass m, you can use two methods. The first method is to consider the C.M. as a semi-circular shell, taking the limit for the radius of the smaller disk to approach the bigger one. The second method is to consider the semi-circular disk as many triangular laminar, where the centroids of these triangles will form an arc that is the C.M. of the disk
  • #1
jessicaw
56
0
i have calculated the C.M. of semi-circular arc of radius r and mass m.
How can i use this answer to calculate the C.M of semi-circular disk of radius r and mass m?
thanks:)

ps. how about the converse?
 
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  • #2


just consider the disk is formed with many semi circular arcs

and for the converse case ,

there are 2 methods

1. consider the CM a semi circular shell, ie. a (disk) - (another disk with smaller radius), then take limit for the radius of the small one approach the bigger one

2. consider the semi circular disk as many triangular laminar...
 
  • #3


sr-candy said:
just consider the disk is formed with many semi circular arcs

and for the converse case ,

there are 2 methods

1. consider the CM a semi circular shell, ie. a (disk) - (another disk with smaller radius), then take limit for the radius of the small one approach the bigger one

2. consider the semi circular disk as many triangular laminar...

"just consider the disk is formed with many semi circular arcs" but isn't the arcs different from each other? I thought of this at first but get stuck. How to do this techinically?
"2. consider the semi circular disk as many triangular laminar..."Triangle??confused.Why?and what do you mean by traingular "laminar"?
 
  • #4


jessicaw said:
"just consider the disk is formed with many semi circular arcs" but isn't the arcs different from each other? I thought of this at first but get stuck. How to do this techinically?
"2. consider the semi circular disk as many triangular laminar..."Triangle??confused.Why?and what do you mean by traingular "laminar"?

1. yes the arcs are of different radius, but you have the CM of the arcs, so you can consider the mass of the arcs are all lying on the y axis, all you have to do is calculate the CM of these "CMs"

2. laminar means layer or plane etc. Just think that a semi circle is formed with many sectors, when the sectors become smaller, it will look like many triangle.
 
  • #5


sr-candy said:
2. laminar means layer or plane etc. Just think that a semi circle is formed with many sectors, when the sectors become smaller, it will look like many triangle.

i understand this now, but how can i use this fact to calculate the C.M of arc? The C.M of triangle is the centroid but the centroid is not on the semicircular arc. So how to use this fact?
 
  • #6


jessicaw said:
i understand this now, but how can i use this fact to calculate the C.M of arc? The C.M of triangle is the centroid but the centroid is not on the semicircular arc. So how to use this fact?

Why not? the centroids of these triangles will form an arc
 

1. What is the definition of the center of mass of a semicircular disk?

The center of mass of a semicircular disk is the point at which the disk's mass is evenly distributed in all directions. It can also be defined as the average position of all the points in the disk, taking into account their mass and distance from the origin.

2. How is the center of mass of a semicircular disk calculated from the center of mass of a semicircular arc?

The center of mass of a semicircular disk can be calculated by taking the average of the x and y coordinates of the center of mass of the semicircular arc, weighted by the respective areas of the arc and the remaining half circle. This can be expressed mathematically as:
xcm = (2/3)R and ycm = (4R/3π).

3. What factors affect the location of the center of mass of a semicircular disk?

The center of mass of a semicircular disk is affected by the mass, distribution of mass, and shape of the disk. It is also influenced by external forces such as gravity or friction.

4. How does the center of mass of a semicircular disk differ from that of a full circular disk?

The center of mass of a semicircular disk is located along the diameter of the disk, while the center of mass of a full circular disk is located at its geometrical center. This is due to the differing distributions of mass in the two shapes.

5. Why is the concept of center of mass important in physics?

The concept of center of mass is important in physics because it helps us understand and predict the motion of objects. It is also used in various calculations such as torque, momentum, and stability of structures. Additionally, it provides a simplified way of analyzing complex systems by treating them as a single point instead of considering each individual particle separately.

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