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How to Find the Centre of Mass of a Semicircular Disc | Testbook
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The centre of mass is a point in an object where its entire weight seems to be concentrated. You can think of it as the balancing point of the object. If you tried to balance the object on your finger, it would stay balanced only if your finger was right under the centre of mass. This concept is very useful in physics, especially when we deal with objects that have odd shapes or are made of different parts.
Instead of calculating how every small part of the object moves, we can treat the whole object as if all its mass is located at the centre of mass. This makes solving problems much easier. In this lesson, we’ll learn how to find the centre of mass for a semicircular disc — a shape that’s like a full circle cut in half.
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Steps to Calculate the Centre of Mass of a Semicircular Disc
Assume the mass of the semicircular disc to be M and its radius to be R. The density or mass per unit area of the disc is then given by
σ = M/(πR2/2) = 2M/πR2
Consider a small element in the shape of a ring with radius r and thickness dr. The area of this element can be calculated as
Area = πrdr (since (dr)2 is very small and can be neglected)
The mass of this elementary ring, dM, can be calculated as
dM = (πrdr)(2M/πR2) = (2Mr/R2)dr
Let's denote the coordinates of the centre of mass of this element as (x,y). Due to the symmetry of the disc, x = 0 and y = 2r/π.
Now, let's denote the coordinates of the centre of mass of the semicircular disc as xcm and ycm.
From symmetry, we can conclude that xcm = 0. The y-coordinate can be calculated as
ycm = (1/M)∫ydM
Substituting the value of y and dM into the equation, we get
After solving the integral, we find
yCM = (4/πR2)(R3/3) = 4R/3π
In conclusion, the centre of mass of a semicircular disc of radius R and mass M is located at the coordinates (0, 4R/3π).
Properties of Centre of Mass of a Semicircular Disc
1. Symmetry of the Disc
A semicircular disc is shaped like half of a full circle. If you draw a vertical line down from the middle of the flat edge (diameter), you will notice that both sides of the disc are mirror images of each other. This vertical line is called the y-axis.
Because of this symmetry, the centre of mass must lie somewhere on this y-axis. This is because the left and right parts of the disc are equal, so the centre of mass can’t be off to one side.
2. Centre of Mass Has No x-Coordinate
Since the shape is symmetrical from left to right, we don’t need to worry about the x-direction (horizontal direction). The x-coordinate of the centre of mass is always zero. So we only need to find its vertical position (the y-coordinate) along the y-axis.
3. It Lies Inside the Semicircle
The centre of mass of a semicircular disc is not on the flat edge (diameter) and not at the centre of the full circle either. Instead, it lies a little below the centre of the diameter, inside the curved area of the disc.
So if you tried to balance the disc at this point, it would stay balanced. This is because it lies within the body of the disc, not outside or on the edge.
4. Depends on Radius (R)
The exact position of the centre of mass along the y-axis depends on the size of the disc — in other words, its radius (R).
Mathematically, this position is found using a special formula:
y = 4R ÷ (3π)
This means the centre of mass is closer to the flat edge than to the curved edge, and the bigger the radius, the farther down the centre of mass lies.
FAQs For Centre of Mass of a Semicircular
What is the Centre of Mass?
Centre of Mass is a fixed point on the object about which the entire mass of the system is equally distributed.
How to calculate the Centre of Mass of a semicircular disc?
The article provides a detailed step-by-step process to calculate the Centre of Mass of a semicircular disc.
Why does the centre of mass lie below the center of the semicircle?
The mass of the disc is distributed more heavily near the base (diameter), so the centre of mass shifts downward from the geometric center of the circle.
Is the centre of mass the same as the centroid for a semicircular disc?
Yes, for a uniform semicircular disc, the centre of mass and the centroid are the same since the mass is uniformly distributed.
How does the centre of mass change for a semicircular ring instead of a disc?
For a semicircular ring (only the boundary, not a filled-in disc), the centre of mass is located at a height of 2R/π above the flat side, which is different from the disc due to different mass distribution.
What is the importance of knowing the centre of mass of a semicircular disc?
It is crucial in mechanics and engineering for analyzing balance, stability, and motion of bodies shaped like or including semicircular sections.
What assumptions are made when calculating the centre of mass of a semicircular disc?
It is assumed that the disc is uniform (constant density and thickness), lies in a 2D plane, and has no external forces acting on it other than gravity.
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