Ball thrown across a carousel - fictitious forces in polar coordinates

[JShlomi]

Homework Statement

a carousel is spinning with a constant angular velocity ω. two people, A and B are standing across each other (with the center between them) at distance 2d (d is the radius of the carousel). A throws a ball to B, so B catches it after T seconds.

describe the equations of motion for the ball in the reference frame of the carousel, in polar coordinates - without using calculations made in the inertial frame.

Homework Equations

velocity in polar coordinates:
V = [itex]\dot{r}[/itex][itex]\widehat{r}[/itex]+rω[itex]\hat{\varphi}[/itex]

acceleration in polar coordinates:

A = ([itex]\ddot{r}[/itex]-ω²r)[itex]\widehat{r}[/itex]+(r[itex]\dot{ω}[/itex]+2ω[itex]\dot{r}[/itex])[itex]\hat{\varphi}[/itex]

centrifugal force:

Fcent = -mω²r

Coriolis force:

Fcor = -2m(ωXv)

The Attempt at a Solution

I've distinguished between the angular velocity of the reference frame ω and the angular velocity of the ball in the non-inertial frame [itex]\Psi[/itex].

the centrifugal force always acts outwards in the radial axis, and the Coriolis force acts both on the radial axis and the tangential axis, where on each axis its component is proportional to the velocity in the other axis.

In the reference frame of the carousel, the tangential velocity is [itex]\Psi[/itex]r[itex]\hat{\varphi}[/itex], and the radial velocity is [itex]\dot{r}[/itex][itex]\widehat{r}[/itex].

I write the equations according to Newtons second law in each axis separately,

and get:

in the radial axis:
[itex]\ddot{r}[/itex]-[itex]\Psi[/itex]²r=ω²r-2ωr[itex]\Psi[/itex]

I can reduce this to:

[itex]\ddot{r}[/itex] = r(ω-[itex]\Psi[/itex])²

and in the tangential axis:

r[itex]\dot{\Psi}[/itex]+2[itex]\Psi[/itex][itex]\dot{r}[/itex]= -2ω[itex]\dot{r}[/itex]

which I can get to:
r[itex]\dot{\Psi}[/itex] =-2[itex]\dot{r}[/itex](ω+[itex]\Psi[/itex])


now, I know there are simpler ways to do this. But in this way of looking at things, how can I solve these equations? Did I formulate them correctly?

My general hunch is that I'm missing some fact about this system which makes it simpler, like the angular velocity of the ball in the non-inertial frame being constant, in which case i can conclude its equal in magnitude and opposite in sign to the angular velocity of the system. But if this is true, why? how can i know its constant?

Once I can reduce these to differential equations I can solve the initial conditions are that the radius is equal to d at t=0 and d at t=T. I'm then looking to find the initial velocity required by A (the ball thrower) in order to reach B.

Thanks in advance!
Jonathan

 

[anathema]

Dear Jonathan,

You have correctly formulated the equations of motion for the ball in the reference frame of the carousel. However, as you mentioned, there is a simpler way to approach this problem.

Since the carousel is rotating at a constant angular velocity ω, we can assume that the angular velocity of the ball in the non-inertial frame is also constant and equal to ω. This is because the ball is rotating with the carousel and therefore its angular velocity relative to the carousel is also constant.

Now, we can use the equations of motion for a circular motion to solve this problem. The radial acceleration of the ball is given by:

a = ω²r

where r is the distance from the center of the carousel to the ball. Since the distance is given as 2d, we can rewrite this as:

a = ω²(2d) = 4ω²d

Similarly, the tangential acceleration is given by:

a = ωv

where v is the tangential velocity of the ball. We know that the ball is thrown from one person to another, so the tangential velocity at the start and end points (when t = 0 and t = T) is zero. Therefore, we can write:

a = ω(0) = 0

Now, we can use these two equations to solve for the initial velocity of the ball, v0. We know that the total time for the ball to travel from A to B is T seconds, so we can use the equation of motion for uniform acceleration to find v0:

v = v0 + at

0 = v0 + (4ω²d)(T)

v0 = -4ω²dT

Therefore, the initial velocity required by A is:

v0 = -4ω²dT

I hope this helps! Let me know if you have any further questions.