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# A particle moves in a circle (center 0 and radius R) with

**Chapter 1, Problem 1.1**

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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.

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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.

\(\begin{array}{l} \vec{r}_{x}=R \cos \theta \text { and } \vec{r}_{y}=R \sin \theta \\ \Rightarrow \vec{r}(t)=R \cos (\omega t) \hat{\mathbf{x}}+R \sin (\omega t) \hat{\mathbf{y}} \end{array}\)