The problem describes a composite object made of four circular rings A, B, C, and D, with radii $r$, $2r$, $2r$, and $3r$, respectively. The task is to find the distance from point X to the center of gravity of the composite object.

1. Problem Description

The problem describes a composite object made of four circular rings A, B, C, and D, with radii $r$, $2r$, $2r$, and $3r$, respectively. The task is to find the distance from point X to the center of gravity of the composite object.

2. Solution Steps

Let's assume the mass of each ring is proportional to its radius. Therefore, the masses of the rings A, B, C, and D are proportional to $r$, $2r$, $2r$, and $3r$, which we can denote as $m$, $2m$, $2m$, and $3m$ respectively.

The center of gravity of the composite object is given by the weighted average of the positions of the centers of the rings. Let the position of X be the origin (0).

The center of ring A is at $r$ from X.

The center of ring B is at $-2r$ from X.

The center of ring C is at $2r$ from X.

The center of ring D is at $-3r$ from X.

The position of the center of gravity ($x_{cg}$) is given by:

$x_{cg} = \frac{m_A x_A + m_B x_B + m_C x_C + m_D x_D}{m_A + m_B + m_C + m_D}$

Substituting the masses and positions:

$x_{cg} = \frac{m(r) + 2m(-2r) + 2m(2r) + 3m(-3r)}{m + 2m + 2m + 3m}$

$x_{cg} = \frac{mr - 4mr + 4mr - 9mr}{8m}$

$x_{cg} = \frac{-8mr}{8m}$

$x_{cg} = -r$

The distance from X is the absolute value of $x_{cg}$, which is $|-r| = r$.

3. Final Answer

The distance from point X to the center of gravity of the composite object is $r$.

(1) $r$