From a unifrom disc of radius R, a circular section of radius (R)/(2) is cut out. The centre of

step 1 My question is from a uniform disk of radius are a circular section of radius R by 2 is cut out.

From a unifrom disc of radius R, a circular section of...

From a unifrom disc of radius $R$, a circular section of radius $\frac{R}{2}$ is cut out. The centre of the hole is at $\frac{R}{2}$ from the centre of the original dise. Locate the centre of gravity of the resulting flat body. (a) $\frac{R}{6}$ to the right of centre $O$ (b) $\frac{R}{3}$ to the right of centre $O$ (c) $\frac{R}{3}$ to the left of centre $O$ (d) $\frac{R}{6}$ to the left of centre 0

From a unifrom disc of radius $R$, a circular section of radius $\frac{R}{2}$ is cut out. The centre of the hole is at $\frac{R}{2}$ from the centre of the original dise. Locate the centre of gravity of the resulting flat body.
(a) $\frac{R}{6}$ to the right of centre $O$
(b) $\frac{R}{3}$ to the right of centre $O$
(c) $\frac{R}{3}$ to the left of centre $O$
(d) $\frac{R}{6}$ to the left of centre 0

Key Concepts

Center of Mass

The center of mass is a point at which the mass of an object can be considered to be concentrated. In mechanics, it is calculated by taking the weighted average of the positions of all mass elements in the body. In composite objects, the center of mass is determined by summing the moments of individual parts and dividing by the total mass.

Composite Bodies and Negative Mass

When dealing with composite or altered objects, such as ones with parts cut out, the removed portion can be treated as having negative mass. This method allows the use of superposition; the overall center of mass of the modified object is found by considering the contributions from each piece (or hole) using the standard formula for center of mass.

Moment of Mass and Weighted Averages

The calculation of the center of mass involves using moments, which are products of the mass of each component and the position vector of its center. The overall center of mass is derived by taking the sum of these moments and dividing by the total mass. This technique applies to both simple shapes and complex configurations.

Uniform Density

Uniform density implies that the mass per unit area (or volume) is constant throughout the object. This simplifies the center of mass calculation by making the mass directly proportional to the area (or volume), allowing the use of geometric proportions rather than complex integrations.

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