Question

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 $(z=0 \text { for both). Here } t$ is in seconds: $x_{1}(t)=4 \cos (2 t)$ $y_{1}(t)=4 \sin (2 t)$ $x_{2}(t)=2 \cos \left(3 t-\frac{\pi}{2}\right)$ $y_{2}(t)=2 \sin \left(3 t-\frac{\pi}{2}\right)$ a. Find the radii of the circles of motion of both particles. b. Find the $x$ - and $y$ -coordinates of the center of mass. c. Decide if the center of mass moves in a circle by plotting its trajectory.

          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 $(z=0 \text { for both). Here } t$ is in seconds:
$x_{1}(t)=4 \cos (2 t)$
$y_{1}(t)=4 \sin (2 t)$
$x_{2}(t)=2 \cos \left(3 t-\frac{\pi}{2}\right)$
$y_{2}(t)=2 \sin \left(3 t-\frac{\pi}{2}\right)$
a. Find the radii of the circles of motion of both particles.
b. Find the $x$ - and $y$ -coordinates of the center of mass.
c. Decide if the center of mass moves in a circle by plotting its trajectory.

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Added by Jasmine C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Timothy James

11/28/2022

Step 1

The radii of the circles of motion of both particles can be found by observing the coefficients of the cosine and sine functions in the given equations. For particle 1, the radius $R_1$ is 4 (from the equations $x_1(t)=4 \cos (2 t)$ and $y_1(t)=4 \sin (2 t)$). For Show more…

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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 $(z=0 \text { for both). Here } t$ is in seconds: $x_{1}(t)=4 \cos (2 t)$ $y_{1}(t)=4 \sin (2 t)$ $x_{2}(t)=2 \cos \left(3 t-\frac{\pi}{2}\right)$ $y_{2}(t)=2 \sin \left(3 t-\frac{\pi}{2}\right)$ a. Find the radii of the circles of motion of both particles. b. Find the $x$ - and $y$ -coordinates of the center of mass. c. Decide if the center of mass moves in a circle by plotting its trajectory.

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