Is w1/w2 = √r2/r1 the Correct Derivation for Circular Central Force?

[SwedishFred]

Hi
we are going through cirkular centralforce and I am complete stuck...
I can't find the derivation to why w1/w2=√r2/r1 is correctw1/w2=√r2/r1sorry I am lost...
best regards
Fred

 

[Dick]

Hi
we are going through cirkular centralforce and I am complete stuck...
I can't find the derivation to why w1/w2=√r2/r1 is correctw1/w2=√r2/r1sorry I am lost...
best regards
Fred

Maybe you could explain what those symbols mean?

 

[SwedishFred]

yes of course , sorry
There are two objects that are spinning on a plate and they fall off, and the physical reasoning is that the angular velocity (ω) has to meet the condition
ω1/ω2=√r2/r1

ω1= angular velocity objekt 1
ω2= angular velocity objekt 2
r2= objet 2 position to the center of the plate
r1= objet 1 position to the center of the plate

Best regards

 

[Dick]

yes of course , sorry
There are two objects that are spinning on a plate and they fall off, and the physical reasoning is that the angular velocity (ω) has to meet the condition
ω1/ω2=√r2/r1

ω1= angular velocity objekt 1
ω2= angular velocity objekt 2
r2= objet 2 position to the center of the plate
r1= objet 1 position to the center of the plate

Best regards

You should probably the objects will fall of when the acceleration exceeds that which can be produced by the frictional force holding them on. Do you know that the acceleration is given by the expression $v^2/r$ where $r$ is distance from the center and $v$ is rotational speed? What's the relation between $v$ and angular velocity?

 

[SwedishFred]

I Think that the relation is V=ωr correct? (and the friktionskoefficient is the same for both the objects)

 

[BvU]

That is correct. Now we need an equation to express 'staying on the plate' versus 'flying off' a bit more in physics terms using the variables in our exercise.
The template had an item for that; unfortunately it has disappeared (how ?, strange !). So here is a copy:

Homework Statement

Homework Equations

The Attempt at a Solution

 

[SwedishFred]

Homework Statement

There are two objects that are spinning on a plate and they fall off, and the physical reasoning is that the angular velocity (ω) has to meet the condition
ω1/ω2=√r2/r1

ω1= angular velocity objekt 1
ω2= angular velocity objekt 2
r2= objet 2 position to the center of the plate
r1= objet 1 position to the center of the plate

Homework Equations

ω1/ω2=√r2/r1
V=ωr

The Attempt at a Solution

√r2/r1 * ω1/ω2=0

√((ω1r2)/(ω2r1))=0

hmm this doenst feel right..

 

[Dick]

Homework Statement

There are two objects that are spinning on a plate and they fall off, and the physical reasoning is that the angular velocity (ω) has to meet the condition
ω1/ω2=√r2/r1

ω1= angular velocity objekt 1
ω2= angular velocity objekt 2
r2= objet 2 position to the center of the plate
r1= objet 1 position to the center of the plate

Homework Equations

ω1/ω2=√r2/r1
V=ωr

The Attempt at a Solution

√r2/r1 * ω1/ω2=0

√((ω1r2)/(ω2r1))=0

hmm this doenst feel right..

You haven't really done anything except for make an algebra mistake. Start from the physics. What condition will make an object fall off the plate?

 

[BvU]

Homework Statement

[/B]
Object 1 lies on a spinning plate at a distance $r_1$ from the axis. It flies off at angular speed $\omega_1$ (at lower speeed it is held in orbit by friction)
Object 2 lies on the same plate at a distance $r_2$ from the axis. It flies off at angular speed $\omega_2$.
Friction coefficients are the same for both objects.
Masses of objects may or may not be the same.

Show that $\omega_1/\omega_2=\sqrt{r_2/r_1}$

Homework Equations

Friction force required to stay in circular orbit F = ...
Maximum centripetal acceleration friction force can provide a_{at fly-off speed} =

The Attempt at a Solution

...

 

[SwedishFred]

Homework Equations

Friction force required to stay in circular orbit F = m·v²/r
Maximum centripetal acceleration friction force can provide a at fly-off speed a_c =v²/r

on the right track?

 

[Dick]

Homework Equations

Friction force required to stay in circular orbit F = m·v²/r
Maximum centripetal acceleration friction force can provide a at fly-off speed a_c =v²/r

on the right track?

Right track. Now replace the $v$ with an expression involving angular velocity.

 

[SwedishFred]

The Attempt at a Solution

F = m·v²/r = (m* (ωr)²)/r
ac =v²/r =(ωr)²/r

 

[Dick]

The Attempt at a Solution

F = m·v²/r = (m* (ωr)²)/r
ac =v²/r =(ωr)²/r

Ok, now simplify that expression. $a_c$ doesn't depend on the mass, agree? So for any two objects $a_c$ is the same.

 

[SwedishFred]

agree
(ωr)²/r = 2ω³r²

 

[Dick]

agree
(ωr)²/r = 2ω³r²

I don't recognize the algebra you did you get that. Can you explain?

 

[SwedishFred]

Ohh sorry calculation fault
I don't know the name in English (kvadreringsregler) (a+b)² = a²+2ab+b²

this one is wrong! se next one..

 

[SwedishFred]

or the potens rule a^xb^x=(ab)^x

 

[SwedishFred]

The Attempt at a Solution

I am overseeing the weight i F.
ω²r=ω²r
ω²/ω²=r/r
ω/ω=√r/r

am I right ??
[/B]

 

[Dick]

The Attempt at a Solution

I am overseeing the weight i F.
ω²r=ω²r
ω²/ω²=r/r
ω/ω=√r/r

am I right ??[/B]

Right idea. It would look much better if you'd distinguish the two values of $\omega$ and $r$. Start from $\omega_1^2 r_1 = \omega_2^2 r_2$.

 

[SwedishFred]

The Attempt at a Solution

ω²₁r₁=ω²₂r₂
ω²₁/ω²₂=r₂/r₁
√(ω²₁/ω²₂)=√(r₂/r₁)
ω₁/ω₂=√(r₂/r₁)

I Think i got it right ?!

 

[Dick]

The Attempt at a Solution

ω²₁r₁=ω²₂r₂
ω²₁/ω²₂=r₂/r₁
√(ω²₁/ω²₂)=√(r₂/r₁)
ω₁/ω₂=√(r₂/r₁)

I Think i got it right ?!

Yes, you've got it!

 

[SwedishFred]

Thanks a lot sir!
we will meet again ;-)
best regards
Fredrik