9.6 Center of Mass

Suppose we included the sun in the system. Approximately where would the center of mass of the Earth-moon-sun system be located? (Feel free to actually calculate it.)

Suppose you have a macroscopic salt crystal (that is, a crystal that is large enough to be visible with your unaided eye). It is made up of a huge number of unit cells. Is the center of mass of this crystal necessarily at the geometric center of the crystal?

How would the firework display change in deep space, far away from any source of gravity?

Two particles of masses $m_{1}$ and $m_{2}$ separated by a horizontal distance D are let go from the same height h at different times. Particle 1 starts at $t = 0$, and particle 2 is let go at $t = T$. Find the vertical position of the center of mass at a time before the first particle strikes the ground. Assume no air resistance.

Two particles of masses $m_{1}$ and $m_{2}$ move uniformly in different circles of radii $R_{1}$ and$R_{2}$ about the origin in the x, y-plane. The coordinates of the two particles in meters are given as follows ($z = 0$ for both). Here t is in seconds:
$\begin{aligned}x_{1} \left(t\right) & = & 4 \text{cos} \left(2 t\right) \\ y_{1} \left(t\right) & = & 4 \text{sin} \left(2 t\right) \\ x_{2} \left(t\right) & = & 2 \text{cos} \left(3 t - \frac{\pi}{2}\right) \\ y_{2} \left(t\right) & = & 2 \text{sin} \left(3 t - \frac{\pi}{2}\right)\end{aligned}$

1. Find the radii of the circles of motion of both particles.
2. Find the x- and y-coordinates of the center of mass.
3. Decide if the center of mass moves in a circle by plotting it…

Find the center of mass of a rod of length L whose mass density changes from one end to the other quadratically. That is, if the rod is laid out along the x-axis with one end at the origin and the other end at $x = L$, the density is given by $\rho \left(x\right) = \rho_{0} + \left(\rho_{1} - \rho_{0}\right) \left(\frac{x}{L}\right)^{2}$, where $\rho_{0}$ and $\rho_{1}$ are constant values.

Find the center of mass of a rectangular material of length a and width b made up of a material of nonuniform density. The density is such that when the rectangle is placed in the xy-plane, the density is given by $\rho \left(x , y\right) = \rho_{0} x y$.

Find the center of mass of a cone of uniform density that has a radius R at the base, height h, and mass M. Let the origin be at the center of the base of the cone and have +z going through the cone vertex.

Find the center of mass of a uniform thin semicircular plate of radius R. Let the origin be at the center of the semicircle, the plate arc from the +x axis to the −x axis, and the z axis be perpendicular to the plate.