a particles location is given by s = 3cos 2ti + 3sin 2tj, confirm its in a circle

[Bronn]

"A particle is moving such that its position is given by:

s = 3cos 2ti + 3sin 2tj

a) confirm the particle is traveling in a circle."



Hey, Ive been really stuck on this problem and Im not really even sure where to get started. My first thoughts were trying to express the equation in some form of x^2 + y^2 = s^2 but i didn't get anywhere with that.
So im pretty stuck, if anyone could atleast give me some clues?
the following questions b c d are simple derivatives and second derivatives finding velocity and acceleration in forms of t. So i would think this question is simple too, if i knew it...

Ive done only some very basic vector stuff, so seeing the i-j hat in with theta is confusing me also.

im having trouble even googling how to do this stuff, so if you know a good video or webpage also would be helpful

thanks

 

[barrick]

"A particle is moving such that its position is given by:

s = 3cos 2ti + 3sin 2tj

a) confirm the particle is traveling in a circle."



Hey, Ive been really stuck on this problem and Im not really even sure where to get started. My first thoughts were trying to express the equation in some form of x^2 + y^2 = s^2 but i didn't get anywhere with that.
So im pretty stuck, if anyone could atleast give me some clues?
the following questions b c d are simple derivatives and second derivatives finding velocity and acceleration in forms of t. So i would think this question is simple too, if i knew it...

Ive done only some very basic vector stuff, so seeing the i-j hat in with theta is confusing me also.

im having trouble even googling how to do this stuff, so if you know a good video or webpage also would be helpful

thanks

Hi Bronn,

The coordinates of the point are:

\(\displaystyle x=3\cos2t\)
\(\displaystyle y=3\sin2t\)
​

Can you compute \(\displaystyle x^2 + y^2\) ?

 

[Bronn]

Hi Bronn,

The coordinates of the point are:

\(\displaystyle x=3\cos2t\)
\(\displaystyle y=3\sin2t\)
​

Can you compute \(\displaystyle x^2 + y^2\) ?

im not sure, you mean something like:

(3 cos 2ti)^2 + (3 sin 2tj)^2

9 cos^2 (2ti) + 9 sin^2 (2tj) = s^2




side question: whats the difference between cos (2ti) and (cos (2t))i ?

 

[barrick]

im not sure, you mean something like:

(3 cos 2ti)^2 + (3 sin 2tj)^2

9 cos^2 (2ti) + 9 sin^2 (2tj) = s^2




side question: whats the difference between cos (2ti) and (cos (2t))i ?

Hi Bronn,

I think there is a problem with the notations: what you have here is a vector equation.

i and j are the unit vectors in the x and y direction, respectively; s is the position vector, whose coordinates are \(\displaystyle x=3\cos2t\) and \(\displaystyle y = 3\sin2t\).

By Pythagoras' theorem, the length of s (which is the distance of the point from the origin) is given by:

\(\displaystyle
\begin{align*}
||\mathbf{s}|| &= \sqrt{x^2 + y^2}\\
&= \sqrt{(3\cos2t)^2 + (3\sin2t)^2}
\end{align*}
\)
​

 

[Bronn]

Hi Bronn,

I think there is a problem with the notations: what you have here is a vector equation.

i and j are the unit vectors in the x and y direction, respectively; s is the position vector, whose coordinates are \(\displaystyle x=3\cos2t\) and \(\displaystyle y = 3\sin2t\).

By Pythagoras' theorem, the length of s (which is the distance of the point from the origin) is given by:

\(\displaystyle
\begin{align*}
||\mathbf{s}|| &= \sqrt{x^2 + y^2}\\
&= \sqrt{(3\cos2t)^2 + (3\sin2t)^2}
\end{align*}
\)
​

i see, thanks. So the distance of s is the radius of the circle?

so with the unit vectors i,j,k does the placement where the i etc goes matter?, as long as its touching the 3cos 2t somewhere?

 

[barrick]

i see, thanks. So the distance of s is the radius of the circle?

so with the unit vectors i,j,k does the placement where the i etc goes matter?, as long as its touching the 3cos 2t somewhere?

Hi Bronn,

Yes. More precisely, the length of the vector s is the distance between the position of the particle and the origin.

This length is constant (=3), as you can see by using a familiar trigonometric identity. This implies that the particle moves on a circle of radius 3.

As far as notations are concerned, the important thing is to clearly separate vectors (like i or j) from coefficients (like 3 cos 2t), to avoid confusion. I think that the best way to achieve this is to use parentheses like (3 cos 2t) i or i (3 cos 2t). It is also customary to use a different font (bold or italic) for vectors when possible.