Consider Fig. 8-14. Three masses located in the x-y plane...

Consider Fig. 8-14. Three masses located in the $x-y$ plane have the following coordinates; a $5 \mathrm{~kg}$ mass has coordinates given by $(2,-3) \mathrm{m} ;$ a $4 \mathrm{~kg}$ mass has coordinates $(-4,2) \mathrm{m} ; \mathrm{a}$ $2 \mathrm{~kg}$ mass has coordinates $(3,3) \mathrm{m}$. Find the coordinates of the center of mass to two significant figures.

Key Concepts

Significant Figures

Significant figures are the digits in a number that contribute to its precision. In applied problems, the final result is often rounded to a specified number of significant figures to communicate the level of precision determined by the input data.

Moment

A moment in mechanics is the product of a mass and its distance from a reference point or axis. When computing the center of mass, the moments of the individual masses about the coordinate axes are summed and divided by the total mass, thereby weighting the contributions of each mass's position.

Coordinate Geometry

Coordinate geometry involves representing points in space using numerical coordinates. In problems like finding the center of mass, coordinates are used to denote positions in the plane, and arithmetic operations on these coordinates establish the location of the center of mass.

Weighted Average

A weighted average is a mean where each component contributes proportionally to its associated weight. In the context of center of mass, each position coordinate is multiplied by its corresponding mass, and this product is summed and then divided by the total mass, resulting in the mass-weighted average positions.

Center of Mass

The center of mass is the point where the entire mass of a system can be considered to be concentrated. It is found by taking the weighted average of the locations of all mass elements in the system, providing a useful simplification when analyzing motion and forces.

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