Object moving on a spiral figure

[Smouk]

Homework Statement

We've got an object/person in the center of a cicrcle that spins around with an angular velocity ω. That same object is moving at a constant speed k with the direction of the radius, that is from the center to the outside of the sphere. That object describes then a spiral movement. We are asked to calculate speed depending on time and also accelaration all done in polar coordinates.

Homework Equations

-We have:
r→=r*Ur→
r=kt
φ=ωt
dr→/dt=v→
dv→/dt=a→

A "→" character means that the previous symbol is a vector.

The Attempt at a Solution

v→ = dr→/dt = dr/dt * Ur→ + r * dUr→/dt = k * Ur→ + rωUφ→ = k * Ur→ + ktωUφ→

a→ = dv→/dt = 0 + kωUφ→ + kωUφ→ + ktω²Ur→ = 2kωUφ→ + ktω²Ur→ =
= 2kωUφ→ + rω²Ur→

The first part (2kωUφ→) would refer to tangential acceleration while the second part (rω²Ur→) would refer to radial acceleration.

Are the answers correct? If not please tell me where I'm wrong and why and if you can also give me some suggestions on presenting problems on a forum like this one.

 

[Simon Bridge]

We've got an object/person in the center of a circle that spins around with an angular velocity ω. That same object is moving at a constant speed k with the direction of the radius, that is from the center to the outside of the sphere.

1. Which is it, a sphere or a circle?
2. Bold face would normally indicate a vector ... angular velocity would be a pseudovector so that's fine, is k a speed or a velocity?

That object describes then a spiral movement. We are asked to calculate speed depending on time and also acceleration all done in polar coordinates.

You seem to be doing work involving a change in reference frame.
It will help you evaluate your results if you are more pedantic about saying which reference frame you are doing what in.
It can also help you check your results by having more than one way to do the problem,
I'll illustrate:

... let's say the "object" is a pen moving in a straight line outwards from the center of a rotating disk so it draws a line on the disk.
That line will be a spiral.

Is that the idea?

The problem, then, would be to work out the speed that the point of the pen moves across the disk.

If the radial speed is k, then the radial position of the point is r'=r=kt.
In the first frame the angular position is always the same, so choose $$\theta^\prime=0$$, and let the reference frames.
Notice I am using the prime to indicate the first reference frame, the rotating frame is unprimed.

In the rotating frame, the point has a tangential component to the velocity: $$v_\perp = r\omega = k\omega t$$

The speed in the rotating frame is the magnitude of the velocity... which is $$|\vec v|^2 = v_r^2 + v_\perp^2$$
... which will let you check your results.

You can use a similar approach to work out acceleration.
I'm guessing that your task is actually to show that you know how to use the polar form of the differentiation and magnitude calculations.
What I'm doing here is showing you how to check.