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What is the position vector r(t) as a function of angle Θ(t). Give your answer about R, Θ(t), and the unit vectors x and y corresponding to the coordinate system.

  1. Find $\theta(t)$ at an arbitrary time t for uniform circular motion. Present the answer in terms of $\omega$ and t.
  2. Find position vector r at time. Present the answer in terms of $R$ and unit vectors x and y.
  3. 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)

Position 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:

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

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