MHB Position of Particle in a Circular Path: Solution

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The discussion focuses on the mathematical representation of a particle's position in circular motion, expressed as r(t) = R(cos(ωt)î + sin(ωt)ĵ). Participants confirm that at t = 0, the particle's position is indeed (R, 0) on the x-axis. They explore the parametrization of the motion through x(t) = Rcos(ωt) and y(t) = Rsin(ωt), leading to the equation x(t)² + y(t)² = R², which confirms the circular path. The conversation also touches on the notation involving unit vectors, clarifying that the vectors are orthogonal and confirming their status as unit vectors. Overall, the thread effectively addresses the mathematical foundations of circular motion and notation used in physics.
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Relatively easy question but I haven't done anything like this in awhile.
$$
\mathbf{r}(t) = \hat{\mathbf{x}}R\cos(\omega t) + \hat{\mathbf{y}}R\sin(\omega t)
$$
The particle moves in a circle so I want to show that the position is given by the above.
I know at $t = 0$ the particle is on the x-axis.
Therefore, we know that at t = 0, $(R,0)$.
Obviously it has to be that but I guess there is a way to show this basically.
 
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re: Particles position

Consider the parametrization:

$\displaystyle x(t)=R\cos(\omega t)$

$\displaystyle y(t)=R\sin(\omega t)$

Square both and add to eliminate the parameter.
 
re: Particles position

MarkFL said:
Consider the parametrization:

$\displaystyle x(t)=R\cos(\omega t)$

$\displaystyle y(t)=R\sin(\omega t)$

Square both and add to eliminate the parameter.

That will just be $x(t)^2 + y(t)^2 = R^2$.
What about x and y hat?
 
re: Particles position

Aren't they orthogonal unit vectors? I'm sorry, I'm not familiar with that notation. I interpreted it as:

$\displaystyle \vec{r}(t)=R\langle \cos(\omega t),\sin(\omega t) \rangle$
 
re: Particles position

MarkFL said:
Aren't they orthogonal unit vectors? I'm sorry, I'm not familiar with that notation. I interpreted it as:

$\displaystyle \vec{r}(t)=R\langle \cos(\omega t),\sin(\omega t) \rangle$

They could be but it doesn't specify. All we know is that they are vectors.
 
re: Particles position

dwsmith said:
They could be but it doesn't specify. All we know is that they are vectors.

I just looked through the book. They are unit vectors. What a dumb notation.
 
re: Particles position

Good deal. I believe I have seen that notation a long time ago in a galaxy far away, but I wasn't sure. ;)
 
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