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edited Sep 4 at 14:56

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You should start thinking of a motion in 2D as a vector quantity.

Given a point taken as a origin $O$, position vector of point $P$ can be thought as the vector going from the origin to the point $P$, $\mathbf{r}_P(t)$.

You can represent the position of the point $P$ in the 2-dimensional plane using different sets of coordinates. As an example:

Cartesian coordinates, $x$, $y$;
polar coordinates, $r$, $\theta$, representing the distance from the origin and the angle measured w.r.t. a reference direction;

Using both coordinates, the position of point $P$ can be written as

$$\begin{aligned} \mathbf{r}_P(t) & = R \cos \omega t \, \mathbf{\hat{x}} + R \sin \omega t \, \mathbf{\hat{y}} = \\ & = R \, \mathbf{\hat{r}}(t) \end{aligned} \ ,$$

being the unit vectors of the Cartesian coordinates $\mathbf{\hat{x}}$, $\mathbf{\hat{y}}$ constant, while the unit vectors of the polar coordinates $\mathbf{\hat{r}}(t)$, $\boldsymbol{\hat{\theta}}(t)$ time-dependent as they follow the motion of the point.

What is constant

For the particular case of uniform circular motion,

the radial component is constant
the tangential component of the velocity is constant $v_{\theta} = \mathbf{v} \cdot \boldsymbol{\hat{\theta}} = \omega R$ is constant
the angular velocity $\boldsymbol{\omega}$

What is not constant

the Cartesian coordinates are not constant $x(t) = R \cos \omega t$, $y(t) = R \sin \omega t$
the velocity is not constant, $$\mathbf{v}(t) = R \omega \boldsymbol{\hat{\theta}}(t) = - R \omega \sin \omega t \mathbf{\hat{x}} + R \omega \cos \omega t \mathbf{\hat{y}}$$

You should start thinking of a motion in 2D as a vector quantity.

Given a point taken as a origin $O$, position vector of point $P$ can be thought as the vector going from the origin to the point $P$, $\mathbf{r}_P(t)$.

You can represent the position of the point $P$ in the 2-dimensional plane using different sets of coordinates. As an example:

Cartesian coordinates, $x$, $y$;
polar coordinates, $r$, $\theta$, representing the distance from the origin and the angle measured w.r.t. a reference direction;

Using both coordinates, the position of point $P$ can be written as

$$\begin{aligned} \mathbf{r}_P(t) & = R \cos \omega t \, \mathbf{\hat{x}} + R \sin \omega t \, \mathbf{\hat{y}} = \\ & = R \, \mathbf{\hat{r}}(t) \end{aligned} \ ,$$

What is constant

For the particular case of circular motion,

the radial component is constant
the tangential component of the velocity is constant $v_{\theta} = \mathbf{v} \cdot \boldsymbol{\hat{\theta}} = \omega R$ is constant
the angular velocity $\boldsymbol{\omega}$

What is not constant

the Cartesian coordinates are not constant $x(t) = R \cos \omega t$, $y(t) = R \sin \omega t$
the velocity is not constant, $$\mathbf{v}(t) = R \omega \boldsymbol{\hat{\theta}}(t) = - R \omega \sin \omega t \mathbf{\hat{x}} + R \omega \cos \omega t \mathbf{\hat{y}}$$

You should start thinking of a motion in 2D as a vector quantity.

Given a point taken as a origin $O$, position vector of point $P$ can be thought as the vector going from the origin to the point $P$, $\mathbf{r}_P(t)$.

You can represent the position of the point $P$ in the 2-dimensional plane using different sets of coordinates. As an example:

Cartesian coordinates, $x$, $y$;
polar coordinates, $r$, $\theta$, representing the distance from the origin and the angle measured w.r.t. a reference direction;

Using both coordinates, the position of point $P$ can be written as

$$\begin{aligned} \mathbf{r}_P(t) & = R \cos \omega t \, \mathbf{\hat{x}} + R \sin \omega t \, \mathbf{\hat{y}} = \\ & = R \, \mathbf{\hat{r}}(t) \end{aligned} \ ,$$

What is constant

For the particular case of uniform circular motion,

the radial component is constant
the tangential component of the velocity is constant $v_{\theta} = \mathbf{v} \cdot \boldsymbol{\hat{\theta}} = \omega R$ is constant
the angular velocity $\boldsymbol{\omega}$

What is not constant

the Cartesian coordinates are not constant $x(t) = R \cos \omega t$, $y(t) = R \sin \omega t$
the velocity is not constant, $$\mathbf{v}(t) = R \omega \boldsymbol{\hat{\theta}}(t) = - R \omega \sin \omega t \mathbf{\hat{x}} + R \omega \cos \omega t \mathbf{\hat{y}}$$

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created Sep 4 at 14:12

basics

You should start thinking of a motion in 2D as a vector quantity.

Given a point taken as a origin $O$, position vector of point $P$ can be thought as the vector going from the origin to the point $P$, $\mathbf{r}_P(t)$.

You can represent the position of the point $P$ in the 2-dimensional plane using different sets of coordinates. As an example:

Cartesian coordinates, $x$, $y$;
polar coordinates, $r$, $\theta$, representing the distance from the origin and the angle measured w.r.t. a reference direction;

Using both coordinates, the position of point $P$ can be written as

$$\begin{aligned} \mathbf{r}_P(t) & = R \cos \omega t \, \mathbf{\hat{x}} + R \sin \omega t \, \mathbf{\hat{y}} = \\ & = R \, \mathbf{\hat{r}}(t) \end{aligned} \ ,$$

What is constant

For the particular case of circular motion,

the radial component is constant
the tangential component of the velocity is constant $v_{\theta} = \mathbf{v} \cdot \boldsymbol{\hat{\theta}} = \omega R$ is constant
the angular velocity $\boldsymbol{\omega}$

What is not constant

the Cartesian coordinates are not constant $x(t) = R \cos \omega t$, $y(t) = R \sin \omega t$
the velocity is not constant, $$\mathbf{v}(t) = R \omega \boldsymbol{\hat{\theta}}(t) = - R \omega \sin \omega t \mathbf{\hat{x}} + R \omega \cos \omega t \mathbf{\hat{y}}$$