Systems and Center of Mass - AP Physics 1

Divide into simple shapes, find each center, and use weighted average. Standard approach for irregular or complex geometries.

At the intersection of the diagonals. Symmetry makes center of mass coincide with geometric center.

It is at the geometric center of the object. Symmetry ensures equal mass distribution around center.

$(x_{cm}, y_{cm}) = (1.875, 2.5)$. Using formula: $x_{cm} = \frac{3(3) + 5(0)}{8} = 1.875$, $y_{cm} = \frac{3(0) + 5(4)}{8} = 2.5$

At the intersection of the diagonals. Symmetry makes center of mass coincide with geometric center.

They do not affect its position; only its motion. External forces change motion but not the relative mass distribution.

$(x_{cm}, y_{cm}) = (4.8, 4.8)$. Using formula: $\frac{4(2) + 6(6)}{10} = 4.8$ for both coordinates.

Mass distribution and positions of the masses. These are the only factors determining center of mass location.

The point where the total mass of a system is considered to be concentrated. Useful for analyzing motion as if all mass is at this single point.

Simplifies calculations by considering motion at this point. Center of mass motion represents overall system behavior.

At the geometric center of the sphere. Spherical symmetry places center at geometric center.

Internal forces do not affect the center of mass. Internal forces are between parts of the system.

$(x_{cm}, y_{cm}) = (7.4, 0)$. Calculation: $\frac{2(1) + 8(9)}{2 + 8} = 7.4$.

Divide into simple shapes, find each center, and use weighted average. Standard approach for irregular or complex geometries.

$x_{cm} = 6$. Calculation: $\frac{2(2) + 4(8)}{2 + 4} = 6$.

$x_{cm} = 2$. Equal masses make center of mass at midpoint: $\frac{0 + 4}{2} = 2$.

The center of mass shifts according to the mass movement. Center of mass follows the moving mass proportionally.

$y_{cm} = \frac{\sum m_i y_i}{\sum m_i}$. Same formula as x-coordinate but using y-positions.

It remains unchanged. Rotation doesn't change relative mass positions.

At the centroid, intersection of medians. For triangles, centroid and center of mass coincide when uniform.

$\vec{r}_{cm} = \frac{1}{M}\int \vec{r} \ dm$. Integral form for objects with continuous mass distribution.

It accelerates uniformly. Center of mass follows Newton's second law like a point mass.

At the center of the disk. Circular symmetry places center of mass at geometric center.

It simplifies the analysis by treating motion as if all mass is at this point. Allows treating complex systems as point masses.

$z_{cm} = \frac{\sum m_i z_i}{\sum m_i}$. Extension to three-dimensional systems using z-coordinates.

At the intersection of the diagonals. Rectangular symmetry puts center at intersection of diagonals.

It is the axis about which the object rotates. Objects tend to rotate about their center of mass.

Friction does not affect the location of the center of mass. Friction is an external force affecting motion, not position.

$(x_{cm}, y_{cm}) = (2.67, 1.33)$. Three equal masses: $\frac{0 + 4 + 4}{3} = 2.67$, $\frac{0 + 0 + 4}{3} = 1.33$.