What is the velocity of the mass in the lab's frame of reference?

[peripatein]

Hi,

Homework Statement

A horizontal smooth disk of radius R rotates around its axis with constant speed ω. At t=0 a mass m is thrown at speed v₀ (in the lab's frame of reference) towards the center of the disk.
I am asked to write down the velocity vector of the mass in the lab's frame of reference and in the disk's. It is stated that in both cases the origin is at the center of the disk.

Homework Equations

The Attempt at a Solution

Primarily, won't the mass's acceleration in the lab's frame of reference be:
a = -2ω x v' - ω x (ω x r), where |v| = v₀ - ωr?

Won't the mass's velocity in the disk's frame of reference be:
v = [ωr]_θ + [dr/dt]_r?
I could truly use some guidance here. Thanks!

 

[rcgldr]

At t=0 a mass m is thrown at speed v₀ (in the lab's frame of reference) towards the center of the disk. ... won't the mass's acceleration ...

The mass isn't accelerating, it's moving at constant velocity v₀, from the lab's frame of reference.

 

[peripatein]

Okay, so is -2ω x v' - ω x (ω x r)=0, where |v| = v0 - ωr?

 

[haruspex]

Okay, so is -2ω x v' - ω x (ω x r)=0, where |v| = v0 - ωr?

In which frame?

 

[peripatein]

Would it be correct to say that in the lab's reference frame, the velocity of the mass is:
V = [wr]_θ+[wtv_0]r?
Would it be correct to say that in the disk's reference frame, the velocity of the mass is:
V = [v_0]θ

 

[peripatein]

I would really appreciate some feedback on what I think the velocities would be in both reference frames.

 

[haruspex]

Would it be correct to say that in the lab's reference frame, the velocity of the mass is:
V = [wr]_θ+[wtv₀]_r?

Not sure I understand your notation. You're using polar co-ordinates for both frames, right? If so, I guess it's the same r for each, and theta's the same at t=0. The mass comes in along theta=0 in the lab's frame.
Given all that, why does the velocity in the lab's frame involve ω? And what would ωtv₀ be... an angle multiplied by a speed?

 

[peripatein]

So will the velocity, from an inertial frame's pov, simply be wr(t), where r=v_rt?
Won't it then be wv_0*t?

 

[haruspex]

As I understand the statement, the mass starts with speed v₀ towards the origin, along the line θ=0, say. In polar, I guess you'd write that (-v₀, 0). Since the disk is smooth, that won't change.

 

[peripatein]

I am not sure I understand. My book claims that to an inertial observer the mass will be moving at constant speed along the radius, i.e. straight line, whereas from the disk's reference frame it will be moving at a speed equal to v_r + w x r.
Would you disagree?

 

[rcgldr]

Would it be correct to say that in the lab's reference frame, the velocity of the mass is:
V = [wr]_θ+[wtv₀]_r?

My book claims that to an inertial observer the mass will be moving at constant speed along the radius, i.e. straight line

So why do you include

[wtv₀]_r

as one of the terms?

In polar coordinates, dr/dt = v₀, and dθ/dt = 0. θ would be a constant, while r = r₀ + v₀ t.