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A particle moves in a circle (center 0 and radius R) with
Chapter 1, Problem 1.1(choose chapter or problem)
A particle moves in a circle (center O and radius R) with constant angular velocity \(\omega\) counter-clockwise. The circle lies in the xy plane and the particle is on the x axis at time t = 0. Show that the particle's position is given by
\(\mathrm{r}(t)=\hat{\mathrm{x}} R \cos (\omega t)+\hat{\mathrm{y}} R \sin (\omega t)\)
Find the particle's velocity and acceleration. What are the magnitude and direction of the acceleration? Relate your results to well-known properties of uniform circular motion.
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QUESTION:
A particle moves in a circle (center O and radius R) with constant angular velocity \(\omega\) counter-clockwise. The circle lies in the xy plane and the particle is on the x axis at time t = 0. Show that the particle's position is given by
\(\mathrm{r}(t)=\hat{\mathrm{x}} R \cos (\omega t)+\hat{\mathrm{y}} R \sin (\omega t)\)
Find the particle's velocity and acceleration. What are the magnitude and direction of the acceleration? Relate your results to well-known properties of uniform circular motion.
ANSWER:Step 1 of 4
Let's find the \(x\) and \(y\) components of the position vector \(r\) with trigonometry.
Also and we know that \(\theta=\omega t\)
without loss of generality we can set the phase to 0 , which means \(\theta=\omega t\) without adding a phase shift.
\(\begin{array}{l} \vec{r}_{x}=R \cos \theta \text { and } \vec{r}_{y}=R \sin \theta \\ \Rightarrow \vec{r}(t)=R \cos (\omega t) \hat{\mathbf{x}}+R \sin (\omega t) \hat{\mathbf{y}} \end{array}\)