Centripetal Acceleration of two masses when one mass is half a radius away

[anomalocaris]

Homework Statement

Two objects, m1 and m2, both of mass m, are place on a horizontal platform which is rotating at a constant angular velocity. m1 is located at a distance R from the axis of rotation and m2 is located at a R. The centripetal acceleration of mass m1 ____ to the centripetal acceleration of m2.

Homework Equations

a_ci=rω²

The Attempt at a Solution

Okay,the correct answer (according to my homework) should be "less than," but I do not understand why. Here's what I did:

Mass 1: a_ci=(1/2)Rω²
so (2a_ci)=Rω²
Mass 2: a_ci=Rω²

So shouldn't mass 1 have more centripetal acceleration? Or should I not have brought the (1/2) to the other side of the equation. My professor gives a lot of questions like this, and I seem to always get them wrong because I make them more complicated than they should be. But it makes sense that an object farther away from the center of rotation should have more acceleration since it would be traveling a greater distance in a larger circle.

 

[tiny-tim]

hi anomalocaris!

Two objects, m1 and m2, both of mass m, are place on a horizontal platform which is rotating at a constant angular velocity. m1 is located at a distance R from the axis of rotation and m2 is located at a R. The centripetal acceleration of mass m1 ____ to the centripetal acceleration of m2.
…
Mass 1: a_ci=(1/2)Rω²
so (2a_ci)=Rω²
Mass 2: a_ci=Rω²

So shouldn't mass 1 have more centripetal acceleration? Or should I not have brought the (1/2) to the other side of the equation. My professor gives a lot of questions like this, and I seem to always get them wrong because I make them more complicated than they should be. But it makes sense that an object farther away from the center of rotation should have more acceleration since it would be traveling a greater distance in a larger circle.

(i take it you mean m₂ is located at 2 R ?)

yes, your common-sense is correct

i honestly don't see how you got the opposite result, even from those equations

but anyway i strongly recommend that you don't use the same letter for two different things …

in this case, call the accelerations a₁ and a₂ (and not both a_ci), and then you can put them both into the same equation, and compare them!

 

[anomalocaris]

Okay! Thanks tiny-tim! So Should I live a1=(1/2)ω^2? Not 2a1=a1=(1/2)ω^2 ? Because that would make a2 the smaller one?

Thank you!

 

[anomalocaris]

hi anomalocaris!


(i take it you mean m₂ is located at 2 R ?)

Oh I see the problem here. I copied and pasted the question, so it left out that m1 is a distance of (1/2)R and m2 is just at R. Sorry for the confusion!

 

[tiny-tim]

hi anomalocaris!

Okay! Thanks tiny-tim! So Should I leave a1=(1/2)ω^2? Not 2a1=a1=(1/2)ω^2 ? Because that would make a2 the smaller one?

Thank you!

it really doesn't matter, so long as you end up with an equation with a₁ on the LHS and a₂ on the RHS

 

[anomalocaris]

Wow embarrassing typographical error! Sometimes I type faster than I think!

Why should they be on opposite sides? How would this look? Sorry for all these questions, I'm probably making a simple concept more complicated, but I just want to understand this since this kind of problem is helpful for solving all kinds of other problems.

 

[tiny-tim]

Why should they be on opposite sides? How would this look?

you're aiming for an equation 2a₁ = a₂

it doesn't matter how you get there (there's more than one way)

anyway, try it and see ​

 

[anomalocaris]

Okay so then a1=(a2/2)? But either way, a1 is greater than a2. The "correct" answer states that a1<a2. Sometimes the HW keys are wrong though.

 

[tiny-tim]

Okay so then a1=(a2/2)? But either way, a1 is greater than a2.

no, a₁=(a₂/2) means a₁ is less than a₂

 

[anomalocaris]

OH! Epiphany moment! Okay I finally get it! a2 is greater because half of it would be equal to one whole a1. Thanks for sticking with me, tiny-tim! I really really appreciate it!