Someone asked me a basic physics question, and I'm not believing my answer, even though it appears to have the correct limiting behavior. This is driving me completely insane.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose I stand on a sidewalk, 10 feet from the middle of an infinitely long and straight road. I have a flashlight in my hand, and I rotate the flashlight at some constant angular speed [tex]\omega[/tex]. The question posed to me was "what is the speed of the light's image along the middle of the road, 30 feet from where I'm standing"?

I'm standing at point O with my flashlight. I place an origin on myself with y pointing "up" and x pointing to the right. So [tex]R=OA=10[/tex]ft and [tex]R'=OB=30[/tex]ft. The velocity of the lightbeam, as a function of position should be:Code (Text):

A B road

* ------------- * ---------------

.

.

*O

[tex]

\vec{v}(\vec{r}) = \omega\,r \left[ \hat{x}\,\cos(\theta) - \hat{y}\, \sin(\theta) \right]

[/tex]

Let [tex]\theta[/tex] be measured from the vertical. At point B,

[tex]

\theta = \cos^{-1}\left(\frac{R}{R'}\right)

[/tex]

And I'm already in trouble. I want the speed of the flashlight along the middle of the road, so I'm asked for [tex]\vec{v}(B)\cdot\hat{x}[/tex]. If you plug in my [tex]\theta[/tex] into my expression for the flashlight's velocity and take the dot product, you get something constant:

[tex]

\vec{v}(B)\cdot\hat{x} = \omega \, R

[/tex]

However, I refuse to believe this. This is saying the flashlight's x component of velocity is constant. My physical intuition says this is nonsense. I can certainly integrate the time it takes for the flashlight to go the entire (infinite) distance of the road. The time *should* be the time it take for me to rotate through [tex]\pi[/tex] radians. That time should be [tex]\pi / \omega[/tex], however, if the x component of velocity is constant, that time will be infinite.

The full answer (what is the velocity of the flashlight anywhere on the middle of the road) is:

[tex]

\vec{v}(R') = \omega \left[\hat{x}\,R - \hat{y} \sqrt{R'^2 - R^2} \right]

[/tex]

which of course becomes [tex]-\infty[/tex] when [tex]R'\rightarrow\infty[/tex] and [tex]\infty[/tex] when [tex]R'\rightarrow-\infty[/tex]. That's expected. The speed of this result is [tex]|\vec{v}(R')| = \omega \, R'[/tex], which is correct also. But how is it possible that the x component remains the same whether [tex]R'\rightarrow\infty[/tex] or [tex]R'\rightarrow R[/tex]. Does that make sense?

I can't see where I went wrong. What's going on here?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Speed of a projection of a rotating object

**Physics Forums | Science Articles, Homework Help, Discussion**