Question

13. Consider three objects, A, B, and C, in R2. Object A is stationary. Object B orbits object A in a circular orbit with a radius of 10. Object C orbits object B in a circular orbit with a radius of 3. Object C moves as a speed such that it makes 12 full revolutions around B for each revolution B makes around A. (a) Find a vector function for the path of motion of object B if object A is placed at the (b) Find a vector function for the path of motion of object C if object B is placed at the (c) Find a vector function for the path of motion of object C if object A is placed at the (d) Use geogebra (the 2d version) or desmos to graph your answer to part (c) and include origin origin origin. this graph with your work.

Answer 1

Assume that at time $t=0$, B and C are on the X axis and that both rotations are counter clockwise.

Angular velocity of B around A is $\omega$. Angular velocity of C around B is $12\omega$

(a)

Vector function of B around A $\vec{r}_{BA}=10(\cos\omega t\hat{i}+\sin\omega t\hat{j})$ .

(b)

Vector function of C around B $\vec{r}_{CB}=3[\cos(12\omega t)\hat{i}+\sin(12\omega t)\hat{j}]$ .

(c)

Vector function of C around A $\vec{r}_{CA}=10[\cos(\omega t)\hat{i}+\sin(\omega t)\hat{j}]+3[\cos(12\omega t)\hat{i}+\sin(12\omega t)\hat{j}]$

$\vec{r}_{CA}=[10\cos(\omega t)+3\cos(12\omega t)]\hat{i}+[10\sin(\omega t)+3\sin(12\omega t)]\hat{j}$

(d)

.