**Find $\theta(t)$ at an arbitrary time t for uniform circular motion. Present the answer in terms of $\omega$ and t.****Find position vector r at time. Present the answer in terms of $R$ and unit vectors x and y.****Find the formula for the position vector of a particle that starts with $ (that\:is, (x_ {0}, y_ {0}) = (0, R)) $ on the positive y axis and then constantly moves in $ \omega $. Show the answer in terms of R, $\omega$ ,t ,and unit vectors x and y.**

The **first part of the question aims** to represent the position vector in terms of $\theta(t)$ and $R$. The **second part of the question seeks** to find $\theta(t)$ for an arbitrary time $t$ for circular motion. The **third part of the question aims** to find position vector $r$ at time $t$. The **last part of the question seeks** to find position vectors in terms of $\omega$, $R$, and $t$.

**Position vectors** are used to indicate the position of a particular body. Knowing the part of the body is essential for explaining the movement of the body. A **position vector** is **a vector** that represents the position or position of any point with respect to a datum such as an origin.** Position vector always** points to a specific topic from the source of this vector. For issues that move along a straight path, the **position vector** that matches the way is most helpful. The **velocity of a point** is equal to the velocity at which the** magnitude of the vector** changes over time, resulting in a vector placed along a line.

## Expert Answer

**Part (1):** **Position vector** $r(t)$ as a **function of angle** $\theta(t)$ in terms of $R$ and $\theta(t)$ is shown as:

\[r(t)=R\cos(\theta t)\vec{i} +R\sin(\theta t)\vec{j}\]

**Part(2):** $\theta(t)$ for **uniform circular motion** at an arbitrary time $t$ in term of $\omega$ and $t$ is shown as:

\[\theta(t)=\omega t\]

**Part(3):** **Position vector** $r(t)$ at** time** $t$ in terms of the $R$ and **position vector** $x$ and $y$.

\[r(t)=R\cos(\omega t)\vec{i}+R\sin(\omega t)\vec{j}\]

**Part(4):** **Position vector** $r$ for a **particle that starts on the positive** $y$ axis and **moves with constant** $\omega$.

\[r=Ri\]

\[r y(t)=-R\sin(\omega t)\vec{i}+R\cos(\omega t)\vec{j}\]

## Numerical Answers

**(1)**

**Position vector** in term of $R$ and $\theta(t)$ is calculated as:

\[r(t)=R\cos(\theta t)\vec{i} +R\sin(\theta t)\vec{j}\]

**(2)**

$\theta$ for **uniform circular motion** at an arbitrary time is shown as:

\[\theta(t)=\omega t\]

**(3)**

Posi**tion vector** $r(t)$ at time $t$ in terms of the $R$ and **position vector** $x$ and $y$ is **calculated** as:

\[r(t)=R\cos(\omega t)\vec{i}+R\sin(\omega t)\vec{j}\]

**(4)**

**Position vector** $r$ for a **particle** is shown as:

\[r=Ri\]

\[r\;y(t)=-R\sin(\omega t)\vec{i}+R\cos(\omega t)\vec{j}\]

## Example

**-What is the position vector $r(t)$ as a function of angle $\theta(t)$. **

**-Find position vector $r$ at time. **

**Solution**

**(a):** **Position vector** $r(t)$ as a** function of angle** $\theta(t)$ in term of $R$ and $\theta(t)$ is **shown** as:

\[r(t)=R\cos(\theta t)\vec{i} +R\sin(\theta t)\vec{j}\]

**(b):** **Position vector** $r(t)$ at **time** $t$ in term of the $\omega$ and $R$ is given as:

\[r(t)=R\cos(\omega t)\vec{i}+R\sin(\omega t)\vec{j}\]