Una partícula P se mueve en una circunferencia. Colocamos un eje de coordenadas XY y en el origen O del sistema de coordenadas en el centro de la circunferencia.
Entonces es
r
→
=
x
r
i
→
+
y
r
j
→
=
(
r
cos
φ
)
i
→
+
(
r
sin
φ
)
j
→
.
{\displaystyle {\overrightarrow {r}}=x_{r}{\overrightarrow {i}}+y_{r}{\overrightarrow {j}}=\left({r\cos \varphi }\right)\,{\overrightarrow {i}}+\left({r\sin \varphi }\right){\overrightarrow {j}}\,.}
ω así
ω
=
lim
Δ
t
→
0
Δ
φ
Δ
t
=
d
φ
d
t
,
{\displaystyle \omega =\lim _{\Delta t\to 0}{\frac {\Delta \varphi }{\Delta t}}={\frac {\operatorname {d} \varphi }{\operatorname {d} t}}\,,}
α
α
=
lim
Δ
t
→
0
Δ
ω
Δ
t
=
d
ω
d
t
=
d
2
φ
d
t
2
.
{\displaystyle \alpha =\lim _{\Delta t\to 0}{\frac {\Delta \omega }{\Delta t}}={\frac {\operatorname {d} \omega }{\operatorname {d} t}}={\frac {\operatorname {d} ^{2}\varphi }{\operatorname {d} t^{2}}}\,.}
t = 0 es también φ = 0, entonces es
φ
(
t
)
=
∫
0
t
ω
d
t
=
∫
0
t
[
∫
0
t
α
d
t
]
d
t
.
{\displaystyle \varphi \left(t\right)=\int _{0}^{t}{\omega \,\operatorname {d} t}=\int _{0}^{t}{\left[{\int _{0}^{t}{\alpha \,\operatorname {d} t}}\right]}\operatorname {d} t\,.}
Un movimiento circular con velocidad angular constante se lo llama uniforme. Entonces
φ
(
t
)
=
φ
(
0
)
+
ω
t
y
p
a
r
a
φ
(
0
)
=
0
⇒
φ
(
t
)
=
ω
t
.
{\displaystyle \varphi (t)=\varphi (0)+\omega t\quad y\;\,para\quad \varphi (0)=0\quad \Rightarrow \quad \varphi (t)=\omega \,t.}
r
→
=
r
(
cos
ω
t
)
i
→
+
r
(
sin
ω
t
)
j
→
.
{\displaystyle {\overrightarrow {r}}=r\left({\cos \omega \,t}\right){\overrightarrow {i}}+r\left({\sin \omega \,t}\right){\overrightarrow {j}}\,.}
v
→
=
d
r
→
d
t
=
−
r
ω
(
sin
ω
t
)
i
→
+
r
ω
(
cos
ω
t
)
j
→
{\displaystyle {\overrightarrow {v}}={\frac {\operatorname {d} {\overrightarrow {r}}}{\operatorname {d} t}}=-r\omega \left({\sin \omega \,t}\right){\overrightarrow {i}}+r\omega \left({\cos \omega \,t}\right){\overrightarrow {j}}}
v
=
v
x
2
+
v
y
2
=
r
ω
sin
2
ω
t
+
cos
2
ω
t
=
r
ω
.
{\displaystyle v={\sqrt {v_{x}^{2}+v_{y}^{2}}}=r\omega {\sqrt {\sin ^{2}\omega t+\cos ^{2}\omega t}}=r\omega \,.}
r y v obtenemos:
r
→
⋅
v
→
=
r
(
cos
ω
t
)
[
−
r
ω
(
sin
ω
t
)
]
+
r
(
sin
ω
t
)
r
ω
(
cos
ω
t
)
=
−
r
2
ω
(
sin
ω
t
)
(
cos
ω
t
)
+
r
2
ω
(
sin
ω
t
)
(
cos
ω
t
)
=
0
{\displaystyle {\begin{aligned}{\overrightarrow {r}}\,\cdot {\overrightarrow {v}}&=r\left({\cos \omega \,t}\right)\left[-r\omega \left({\sin \omega \,t}\right)\right]+r\left({\sin \omega \,t}\right)r\omega \left({\cos \omega \,t}\right)\\&=-r^{2}\omega \left({\sin \omega \,t}\right)\left({\cos \omega \,t}\right)+r^{2}\omega \left({\sin \omega \,t}\right)\left({\cos \omega \,t}\right)\\&=0\end{aligned}}}
r y v son perpendiculares . Para la aceleración tenemos que
a
→
=
d
v
→
d
t
=
−
r
ω
2
(
cos
ω
t
)
i
→
−
r
ω
2
(
sin
ω
t
)
j
→
{\displaystyle {\overrightarrow {a}}={\frac {\operatorname {d} {\overrightarrow {v}}}{\operatorname {d} t}}=-\,r\omega ^{2}\left({\cos \omega \,t}\right){\overrightarrow {i}}-r\omega ^{2}\left({\sin \omega \,t}\right){\overrightarrow {j}}}
a
→
=
−
ω
2
r
→
⇒
a
=
ω
2
r
=
v
2
r
.
{\displaystyle {\overrightarrow {a}}=-\,\omega ^{2}{\overrightarrow {r}}\quad \Rightarrow \quad a=\omega ^{2}r={\frac {v^{2}}{r}}\,.}
Aqui la aceleración angular α es constante y también ω (0) = 0
ω
(
t
)
=
α
t
=
(
d
φ
d
t
)
t
.
{\displaystyle \omega \left(t\right)=\alpha \,t=\left({\frac {\operatorname {d} \varphi }{\operatorname {d} t}}\right)_{t}.}
También, cuando φ (0)=0, así para el angulo de rotación
φ
(
t
)
=
∫
0
t
ω
d
t
=
∫
0
t
α
t
d
t
=
α
2
t
2
.
{\displaystyle \varphi \left(t\right)=\int _{0}^{t}{\omega \,\operatorname {d} t}=\int _{0}^{t}{\alpha \,t}\,\operatorname {d} t={\frac {\alpha }{2}}t^{2}.}
r
→
=
r
(
cos
α
2
t
2
)
i
→
+
r
(
sin
α
2
t
2
)
j
→
{\displaystyle {\overrightarrow {r}}=r\left({\cos {\frac {\alpha }{2}}\,t^{2}}\right){\overrightarrow {i}}+r\left({\sin {\frac {\alpha }{2}}\,t^{2}}\right){\overrightarrow {j}}\,}
v
→
=
d
r
→
d
t
=
r
α
t
[
−
(
sin
α
2
t
2
)
i
→
+
(
cos
α
2
t
2
)
j
→
]
=
r
ω
[
−
(
sin
α
2
t
2
)
i
→
+
(
cos
α
2
t
2
)
j
→
]
{\displaystyle {\overrightarrow {v}}={\frac {\operatorname {d} {\overrightarrow {r}}}{\operatorname {d} t}}=r\alpha \,t\left[{-\left({\sin \,{\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {i}}+\left({\cos {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {j}}}\right]=r\omega \left[{-\left({\sin \,{\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {i}}+\left({\cos {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {j}}}\right]}
a
→
=
d
v
→
d
t
=
r
α
[
−
(
sin
α
2
t
2
)
i
→
+
(
cos
α
2
t
2
)
j
→
]
+
{\displaystyle {\overrightarrow {a}}={\frac {\operatorname {d} {\overrightarrow {v}}}{\operatorname {d} t}}=r\alpha \left[{-\left({\sin {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {i}}+\left({\cos {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {j}}}\right]+}
+
r
α
2
t
2
[
(
−
cos
α
2
t
2
)
i
→
−
(
sin
α
2
t
2
)
j
→
]
.
{\displaystyle +\,r\alpha ^{2}t^{2}\left[{\left({-\cos {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {i}}-\left({\sin {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {j}}}\right].}
a
→
=
[
−
r
α
(
sin
φ
)
−
r
ω
2
(
cos
φ
)
]
i
→
+
{\displaystyle {\overrightarrow {a}}=\left[{-r\alpha \left({\sin \varphi }\right)-r\omega ^{2}\left({\cos \varphi }\right)}\right]{\overrightarrow {i}}+}
+
[
r
α
(
cos
φ
)
−
r
ω
2
(
sin
φ
)
]
j
→
.
{\displaystyle +\left[{r\alpha \left({\cos \varphi }\right)-r\omega ^{2}\left({\sin \varphi }\right)}\right]{\overrightarrow {j}}.}
a
r
a
d
=
r
ω
2
{\displaystyle \,a_{rad}=r\omega ^{2}}
a
t
a
n
=
r
α
{\displaystyle \,a_{tan}=r\alpha }
La velocidad angular como medida de direccion
editar
A veces es muy útil ver a la velocidad angular como medida de la direccion y representarlo a través de un en el eje de
![{\displaystyle \varphi \left(t\right)=\int _{0}^{t}{\omega \,\operatorname {d} t}=\int _{0}^{t}{\left[{\int _{0}^{t}{\alpha \,\operatorname {d} t}}\right]}\operatorname {d} t\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f88051e114316d650fb4fd419d2f83eac12c73e1)
![{\displaystyle {\begin{aligned}{\overrightarrow {r}}\,\cdot {\overrightarrow {v}}&=r\left({\cos \omega \,t}\right)\left[-r\omega \left({\sin \omega \,t}\right)\right]+r\left({\sin \omega \,t}\right)r\omega \left({\cos \omega \,t}\right)\\&=-r^{2}\omega \left({\sin \omega \,t}\right)\left({\cos \omega \,t}\right)+r^{2}\omega \left({\sin \omega \,t}\right)\left({\cos \omega \,t}\right)\\&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2452256307901ce1a7b53b4592a6da7b29915a0e)
![{\displaystyle {\overrightarrow {v}}={\frac {\operatorname {d} {\overrightarrow {r}}}{\operatorname {d} t}}=r\alpha \,t\left[{-\left({\sin \,{\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {i}}+\left({\cos {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {j}}}\right]=r\omega \left[{-\left({\sin \,{\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {i}}+\left({\cos {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {j}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/610a2564823b8defefeeecbe611e585aca6af86a)
![{\displaystyle {\overrightarrow {a}}={\frac {\operatorname {d} {\overrightarrow {v}}}{\operatorname {d} t}}=r\alpha \left[{-\left({\sin {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {i}}+\left({\cos {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {j}}}\right]+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f27d7325d8770cf13ee8446c138f086d328e25d8)
![{\displaystyle +\,r\alpha ^{2}t^{2}\left[{\left({-\cos {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {i}}-\left({\sin {\frac {\alpha }{2}}t^{2}}\right){\overrightarrow {j}}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b85262543608f244520a74930287c29b567f7430)
![{\displaystyle {\overrightarrow {a}}=\left[{-r\alpha \left({\sin \varphi }\right)-r\omega ^{2}\left({\cos \varphi }\right)}\right]{\overrightarrow {i}}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5796d2f39e4a3a173a01ac6885e4fd4703696c98)
![{\displaystyle +\left[{r\alpha \left({\cos \varphi }\right)-r\omega ^{2}\left({\sin \varphi }\right)}\right]{\overrightarrow {j}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f61d1f3b5ba7920a08d07a4d4628620cad7e77e3)
