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

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SUMMARY

The derivation of the relationship between angular velocities and radial positions for two objects on a spinning plate is confirmed as ω1/ω2 = √(r2/r1). Here, ω1 and ω2 represent the angular velocities of object 1 and object 2, respectively, while r1 and r2 denote their distances from the center of the plate. The discussion emphasizes the importance of centripetal acceleration and friction in maintaining circular motion, with the maximum centripetal acceleration being expressed as a = v²/r, where v is the linear speed related to angular velocity by v = ωr.

PREREQUISITES
  • Understanding of angular velocity (ω) and its relationship to linear speed (v)
  • Knowledge of centripetal acceleration and its formula a = v²/r
  • Familiarity with the concept of frictional force in circular motion
  • Basic algebraic manipulation skills for deriving equations
NEXT STEPS
  • Study the derivation of centripetal acceleration and its applications in circular motion.
  • Learn about the effects of friction on circular motion and how it influences stability.
  • Explore the relationship between angular velocity and linear velocity in greater detail.
  • Investigate real-world applications of circular motion principles in engineering and physics.
USEFUL FOR

Students of physics, particularly those studying mechanics, as well as educators and anyone interested in the dynamics of circular motion and angular relationships.

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

 

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..
 
SwedishFred said:

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?
 

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 aat fly-off speed =


The Attempt at a Solution



...
 
  • #10

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 ac =v2/r

on the right track?
 
  • #11
SwedishFred said:

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 ac =v2/r

on the right track?

Right track. Now replace the ##v## with an expression involving angular velocity.
 
  • #12

The Attempt at a Solution


F = m·v2/r = (m* (ωr)2)/r
ac =v2/r =(ωr)2/r
 
  • #13
SwedishFred said:

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.
 
  • #14
agree
(ωr)2/r = 2ω3r2
 
  • #15
SwedishFred said:
agree
(ωr)2/r = 2ω3r2

I don't recognize the algebra you did you get that. Can you explain?
 
  • #16
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..
 
Last edited:
  • #17
or the potens rule axbx=(ab)x
 
  • #18

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]
 
  • #19
SwedishFred said:

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##.
 
  • #20

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 ?!
 
  • #21
SwedishFred said:

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!
 
  • Like
Likes   Reactions: SwedishFred
  • #22
Thanks a lot sir!
we will meet again ;-)
best regards
Fredrik
 

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