# Circular motion

## Bidimensional generic motion

Considering a curvilinear trajectory on which we choose an origin ${\displaystyle O}$ and a way of movement:

Definition (Curvilinear abscissa (l))

We define curvilinear abscissa the portion of bend which connects ${\displaystyle x_{1}}$ with ${\displaystyle O}$. If ${\displaystyle x_{1}}$ is among the positive ${\displaystyle x}$, according to the chosen way of movement, the curvilinear abscissa ${\displaystyle Ox_{1}}$ is positive; otherwise if ${\displaystyle x_{1}}$ is among the negative ${\displaystyle x}$, according to the chosen way of movement, the curvilinear abscissa is negative. The velocity of the point defines the concordance of symbol between the chosen way of movement and the way of movement of the point: positive velocity makes the point move according to the chosen way of movement, negative velocity makes the point move in the opposite way.

Average velocity

${\displaystyle V_{m}(t_{1},t_{2})={\frac {x(t_{2})-x(t_{1})}{t_{2}-t_{1}}}}$

is represented by the vector with the same track of ${\displaystyle x_{1}x_{2}}$ and the same direction of motion. But we notice how this definition of average velocity gives very general information.

We use now the definition of derivative to calculate the instantaneous velocity (or just velocity):

${\displaystyle V=\lim _{\Delta t\rightarrow 0}V_{m}(t,t+\Delta t)=\lim _{\Delta t\rightarrow 0}{\frac {\Delta x}{\Delta t}}={\frac {dx}{dt}}}$
whose vector is tangential to the trajectory in ${\displaystyle x_{1}}$ where we find the point at the instant ${\displaystyle t}$ considered.

If we apply a second time the derivative we find the instantaneous acceleration (or just acceleration):

${\displaystyle a=\lim _{\Delta t\rightarrow 0}{\frac {\Delta v}{\Delta t}}={\frac {dv}{dt}}={\frac {d}{dt}}\cdot {\frac {dx}{dt}}={\frac {d^{2}x}{dt^{2}}}}$
whose vector is parallel to the banding ray in ${\displaystyle x_{1}}$, and perpendicular to the vector ${\displaystyle v(t)}$.

## Generic 3D motion

${\displaystyle {\hat {i}},{\hat {j}},{\hat {k}}}$ are versor and they are constant all over the space. ${\displaystyle {\vec {x}}(t)}$ is the position versor. Breaking it up on the axes x, y e z we find:

${\displaystyle {\vec {x}}(t)=x(t)\cdot {\hat {i}}+y(t)\cdot {\hat {j}}+z(t)\cdot {\hat {k}}\ .}$

From the decomposition of ${\displaystyle {\vec {x}}(t)}$ we break up on the three axis ${\displaystyle {\vec {v}}(t)}$:

${\displaystyle {\vec {v}}(t)={\frac {d{\vec {x}}}{dt}}={\frac {dx}{dt}}{\hat {i}}+{\frac {dy}{dt}}{\hat {j}}+{\frac {dz}{dt}}{\hat {k}}\ .}$
We also know that:
${\displaystyle {\vec {v}}(t)=v_{x}{\hat {i}}+v_{y}{\hat {j}}+v_{z}{\hat {k}}\ ,}$
and we find the equivalence:
${\displaystyle v_{x}={\frac {dx}{dt}};\qquad v_{y}={\frac {dy}{dt}};\qquad v_{z}={\frac {dz}{dt}}\ .}$

We can also break up the acceleration vector ${\displaystyle {\vec {a}}(t)}$:

${\displaystyle {\vec {a}}(t)={\frac {d{\vec {v}}}{dt}}={\frac {dv_{x}}{dt}}{\hat {i}}+{\frac {dv_{y}}{dt}}{\hat {j}}+{\frac {dv_{z}}{dt}}{\hat {k}}\ .}$
We also know that:
${\displaystyle {\vec {a}}(t)=a_{x}{\hat {i}}+a_{y}{\hat {j}}+a_{z}{\hat {k}}}$
as we have done for the velocity, we find that:
${\displaystyle a_{x}={\frac {dv_{x}}{dt}}={\frac {d^{2}x}{dt}};\qquad a_{y}={\frac {dv_{y}}{dy}}={\frac {d^{2}y}{dt}};\qquad a_{z}={\frac {dv_{z}}{dt}}={\frac {d^{2}z}{dt}}\ .}$

## Circular motion

Definition (Circular motion)

We define circular motion the specific curvilinear motion in which the trajectory is a circumference of centre ${\displaystyle O}$ and ray ${\displaystyle R}$

Definition (Curvilinear coordinate)

We define curvilinear coordinate (indicated with ${\displaystyle s}$) the oriented length of the arch

The equation of motion is of the form

${\displaystyle \theta =\theta (t)}$

If the circular motion is uniform, the speed is constant: ${\displaystyle \omega =\omega _{0}}$ and

${\displaystyle {\frac {d\theta }{dt}}=\omega _{0};\quad d\theta =\omega _{0}dt;}$
${\displaystyle \int _{\theta _{0}}^{\theta (t)}d\theta =\int _{t_{0}}^{t}\omega _{0}dt;}$
${\displaystyle [\theta ]_{\theta _{0}}^{\theta (t)}=\omega _{0}[t]_{t_{0}}^{t};\quad \theta (t)-\theta _{0}=\omega _{0}t-\omega _{0}t_{0};}$
As ${\displaystyle t_{0}=0s,\quad \omega _{0}t_{0}=0}$, we find:
${\displaystyle \theta (t)=\theta _{0}+\omega _{0}t}$

we now impose ${\displaystyle \theta _{0}=0_{\text{rad}}}$,

${\displaystyle \theta =\omega t}$

• Decomposition of motion:

We now have to apply what said on the decomposition of ${\displaystyle v(t)}$ and ${\displaystyle a(t)}$ in the general curvilinear motion, and we find:

{\displaystyle \left\{{\begin{aligned}&x(t)=r\cos \theta =r\cos(\omega t)\\&y(t)=r\sin(\omega t)\\\end{aligned}}\right.}

{\displaystyle \left\{{\begin{aligned}&V_{x}(t)={\frac {dx}{dt}}=-\omega r\sin(\omega t)\\&V_{y}(t)=\omega r\cos(\omega t)\\\end{aligned}}\right.}

{\displaystyle \left\{{\begin{aligned}&a_{x}(t)={\frac {dV_{x}}{dt}}=-\omega ^{2}r\cos(\omega t)\\&a_{y}(t)={\frac {dV_{y}}{dt}}=-\omega ^{2}r\sin(\omega t)\\\end{aligned}}\right.}

where ${\displaystyle a}$ is called centripetal acceleration

Definition (Centripetal acceleration)

We define centripetal acceleration, the acceleration which causes the bending of trajectory without modifying the absolute value of the speed ω

→ This is why we talk about of uniform motion even though there is an acceleration

• Relation between v, ω, a

${\displaystyle \theta ={\frac {S}{R}}}$

${\displaystyle \omega ={\frac {d\theta }{dt}}={\frac {V}{R}}}$
As ${\displaystyle V={\frac {dS}{dt}}}$ we find that
${\displaystyle \omega ={\frac {1}{R}}\cdot {\frac {dS}{dt}}}$
.

${\displaystyle |{\bar {a}}|={\sqrt {a_{x}^{2}+a_{y}^{2}}}={\sqrt {(\omega ^{2}R\cos(\omega t))^{2}+(\omega ^{2}R\sin(\omega t))^{2}}}=\omega ^{2}R{\sqrt {\cos ^{2}(\omega t)+\sin ^{2}(\omega t)}}}$
As ${\displaystyle \cos ^{2}\alpha +\sin ^{2}\alpha =1}$ we find that ${\displaystyle \cos ^{2}(\omega t)+\sin ^{2}(\omega t)=1}$ so
${\displaystyle |{\bar {a}}|=\omega ^{2}R\cdot 1=\omega ^{2}R}$
We know that ${\displaystyle \omega ={\frac {V}{R}}}$ and we can find the acceleration as a function of velocity
${\displaystyle a_{c}={\frac {V^{2}}{R}}}$