A uniform circular disc of radius a is taken. Λcircular portion of radius b has been removed fro

step 1 Okay, in this question we are asked that a uniform circular disk of radius A is taken. A circula

A uniform circular disc of radius a is taken. Λcircular...

A uniform circular disc of radius $a$ is taken. $\Lambda$ circular portion of radius $b$ has been removed from it as shown in the figure. If the centre of hole is al a distance $c$ from the centre o the disc, the distance $x_{2}$ of the centre of mass of the remaining part from the initial centre of mass $O$ is given by: (a) $\frac{\pi b^{2}}{\left(a^{2}-c^{2}\right)}$ (b) $\frac{c b^{2}}{\left(a^{2}-b^{2}\right)}$ (c) $\frac{\pi c^{2}}{\left(a^{2} b^{2}\right)}$ (d) $\frac{c a^{2}}{\left(c^{2} b^{2}\right)}$

A uniform circular disc of radius $a$ is taken. $\Lambda$ circular portion of radius $b$ has been removed from it as shown in the figure. If the centre of hole is al a distance $c$ from the centre o the disc, the distance $x_{2}$ of the centre of mass of the remaining part from the initial centre of mass $O$ is given by:
(a) $\frac{\pi b^{2}}{\left(a^{2}-c^{2}\right)}$
(b) $\frac{c b^{2}}{\left(a^{2}-b^{2}\right)}$
(c) $\frac{\pi c^{2}}{\left(a^{2} b^{2}\right)}$
(d) $\frac{c a^{2}}{\left(c^{2} b^{2}\right)}$

Key Concepts

Uniform Density

Uniform density implies that the mass per unit area (in two dimensions) or volume (in three dimensions) is constant throughout the object. This allows for a direct proportionality between mass and geometric dimensions like area, making it simpler to calculate contributions to the centre of mass from different regions of the composite object.

Negative Mass Method

The negative mass method is a technique used in calculating the centre of mass of a composite body where a portion of the body has been removed. In this approach, the removed portion is treated as if it possesses a negative mass, and its effect on the overall centre of mass is subtracted from that of the full body. This simplifies the calculation by allowing the use of the standard centre of mass formulas while accounting for the absence of material.

Centre of Mass

The centre of mass of a body is the weighted average location of all its mass points. It is the point where the entire mass of the object can be assumed to be concentrated when considering translational motion. Calculating the centre of mass is crucial in dynamics and statics, especially for bodies with complex or non-uniform mass distributions.

Composite Bodies

Composite bodies refer to objects that are made up of several parts or whose shape can be thought of as a combination (or subtraction) of simpler shapes. In determining the centre of mass for such bodies, each component's mass and its position is considered. The centre of mass of the overall system is found by taking the sum of the moments of the individual parts, sometimes treating subtracted parts as having negative mass.

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