Trajectory in a non inertial frame

[LCSphysicist]

Homework Statement

The hallmark of an inertial reference frame is that any object which is subject to zero net force will travel in a straight line at constant speed. To illustrate this, consider the following experiment: I am standing on the ground (which we shall take to be an inertial frame) beside a perfectly flat horizontal turntable, rotating with constant angular velocity ω. I lean over and shove a frictionless puck so that it slides across the turntable, straight through the center. The puck is subject to zero net force and, as seen from my inertial frame, travels in a straight line. Describe the puck's path as observed by someone sitting at rest on the turntable. This requires careful thought, but you should be able to get a qualitative picture. For a quantitative picture, it helps to use polar coordinates.

Relevant Equations

r = r*ur
v = r*ur + wru(theta)

Well, make the center of the polar coordinates at the center of the turntable, so put r along the ro initial.
I am well known that to someone who is rest on the turntable, the equations will be the follow:

dr/dt = ro - vt

If the turntable route with angular velocity w,

dtheta/dt = -wt

We could go on with this and finish the question.

But i want to know about the other way that i think to resolve the question starting by the velocities:

We will have v = v = r*ur + wr*u(theta)
v = -v*ur - w(ro-vt)u(theta)

I don't know how should i proceed here..

 

[etotheipi]

To make it simpler, you might consider the puck to be released from the centre of the table. If the (polar) coordinates of the puck in the lab frame are $(r, \theta) = (vt, 0)$, then what are the polar coordinates of the puck in the rotating frame? It might help to draw a quick sketch at some time $t$.

Once you have time parameterised $r$ and $\theta$, you can then eliminate $t$. What sort of shape do you get?

 

[LCSphysicist]

To make it simpler, you might consider the puck to be released from the centre of the table. If the (polar) coordinates of the puck in the lab frame are $(r, \theta) = (vt, 0)$, then what are the polar coordinates of the puck in the rotating frame? It might help to draw a quick sketch at some time $t$.

Once you have time parameterised $r$ and $\theta$, you can then eliminate $t$. What sort of shape do you get?

Just to simplify the shape

r = vt
theta = wt

r = v*theta/w

r = K*theta

A type of spiral.

Well, in my question and your example it's easier see the motion by take it directly, but i am confused if i tried to do it starting with the velocity.

Your example:
the minus sign because to someone in rest at the center, he doesn't route, the stone or whatever, that starts a moving this spiral.

v = v*ur ´- wr*utheta
v = v*ur - wvt*utheta

Make sense i try to integrate this? I think not, maybe is this a dead end and have to do the first approach?

 

[LCSphysicist]

 

[etotheipi]

You can try a velocity approach as well. It's perhaps easiest in Cartesian coordinates. In the lab frame, the velocity might be $\vec{v} = v \hat{x}$, or more succinctly $v_x = v$.

You might then try and find what the components of velocity $v_x'$ and $v_y'$ are in a rotating frame (with rotating $x'$ and $y'$ axes.

But be careful: the $\hat{x}'$ and $\hat{y}'$ basis vectors are also functions of time...

 

[jbriggs444]

View attachment 262639

That's half of it.