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![Four beads, each of mass \( m = 1 \) kg, are attached at various locations to a ring, also of mass \( m = 1 \) kg, and radius \( R = 1 \) m (see figure). Find the coordinates of the center of mass of the system consisting of the ring and the beads.
Angle \( A \), located between the first bead and the horizontal, is equal to \( 43^\circ \). Angle \( B \), located between the horizontal and the second bead, is equal to \( 50^\circ \). Angle \( C \), located between the third bead and the horizontal, is equal to \( 68^\circ \). Angle \( D \), located between the horizontal and the fourth bead, is equal to \( 31^\circ \).
\[ x_{cm} = \quad \text{m} \]
\[ y_{cm} = \quad \text{m} \]
**Diagram Explanation:**
- The diagram features a circle representing a ring, centered around the origin of an \( xy \)-coordinate system.
- Four beads, labeled \( m_1, m_2, m_3, \) and \( m_4 \), are positioned at various angles on the circumference.
- Light blue lines connect each bead to the center of the circle, depicting their radial positions.
- Arrows labeled \( A, B, C, \) and \( D \) indicate the angles between the horizontal axis and the lines connecting the center to each bead.
- The \( x \)-axis runs horizontally, and the \( y \)-axis runs vertically, intersecting at the center.](https://content.bartleby.com/qna-images/question/0f6dc858-dc6a-4614-8154-d70dfca22707/9557ed4e-ecdd-4971-a3c0-70c97a8b9364/q12bnn_processed.png)
Transcribed Image Text:Four beads, each of mass \( m = 1 \) kg, are attached at various locations to a ring, also of mass \( m = 1 \) kg, and radius \( R = 1 \) m (see figure). Find the coordinates of the center of mass of the system consisting of the ring and the beads.
Angle \( A \), located between the first bead and the horizontal, is equal to \( 43^\circ \). Angle \( B \), located between the horizontal and the second bead, is equal to \( 50^\circ \). Angle \( C \), located between the third bead and the horizontal, is equal to \( 68^\circ \). Angle \( D \), located between the horizontal and the fourth bead, is equal to \( 31^\circ \).
\[ x_{cm} = \quad \text{m} \]
\[ y_{cm} = \quad \text{m} \]
**Diagram Explanation:**
- The diagram features a circle representing a ring, centered around the origin of an \( xy \)-coordinate system.
- Four beads, labeled \( m_1, m_2, m_3, \) and \( m_4 \), are positioned at various angles on the circumference.
- Light blue lines connect each bead to the center of the circle, depicting their radial positions.
- Arrows labeled \( A, B, C, \) and \( D \) indicate the angles between the horizontal axis and the lines connecting the center to each bead.
- The \( x \)-axis runs horizontally, and the \( y \)-axis runs vertically, intersecting at the center.
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