How Do Fictitious Forces Affect Bead Motion on a Circle in a Non-Inertial Frame?

[peripatein]

Hi,

Homework Statement

I have a question concerning fictitious forces (although its first part is irrelevant to the latter):
A smooth string is extended between two points A and B in a vertically positioned circle, so that the angle between the string and the vertical axis of the circle is denoted β. A bead of mass m slides from rest down a smooth chord in a vertically positioned circle.
(a) I was first asked to show that the time it would take the bead to traverse the distance AB is not dependent on the angle β.
(b) I was then told that the entire set up was put in a non-inertial system, namely a cart accelerating with constant acceleration a' to the right, and was asked to find from what point A' need now the bead to be released so that the duration of slide (time required to traverse A'B') is not dependent on the length of the chord.

Homework Equations

The Attempt at a Solution

(a) Supposing my choice of coordinate system is such that is parallel to the slide (hence, to the string):
mgsin(β) = N ; mgcos(β) = ma; length of string (=L)*cos(β) = 1/2*a*t²
These three equations yielded t = sqrt(2L/g).
Is that correct?
(b) I realize that a fictitious force is now in action, equal to ma', whose direction is to the left. I wrote down the following equations:
N = mgsin(β) + ma'cos(β); mgcos(β) - ma'sin(β) = ma
Are these two correct?
I am not sure how to proceed. Would appreciate some advice.

 

[rcgldr]

Rather than trying to solve the equation for (b) in terms of β, it might be easier to consider what was the condition that made case (a) true in terms of the position A, and how to adjust A' so that a similar condition exists for case (b).

 

[peripatein]

But a similar condition for case (b) would entail dependence on the length of string (supposing the expression I got for case (a) is correct), wouldn't it?

 

[rcgldr]

But a similar condition for case (b) would entail dependence on the length of string (supposing the expression I got for case (a) is correct), wouldn't it?

Did the length of the string matter for case (a)? Wasn't the length of the string different depending on B? What if the strength of gravity was different for case (a), would it still be true that the time would be independent of β? If so, then if the strength of gravity doesn't matter for case (a), then what is important about where A is located (for case (a))?

 

[peripatein]

Do you mean, what made case (a) not dependent on the angle beta?
Furthermore, is the answer I provided for case (a) correct?

 

[rcgldr]

Do you mean, what made case (a) not dependent on the angle beta?

Yes.

Furthermore, is the answer I provided for case (a) correct?

I didn't check it, but the problem states that the time doesn't depend on β, so you can assume this is true for case (a). The position of A for case (a) is important. What is special about the position of A for case(a)? I left a hint in this spoiler:

Spoiler

what is the direction of gravity from position A on the circle?

 

[peripatein]

Let me take you back a bit. I am trying to rewrite the equations for case (a), and do not quite understand how the time it would take the bead to traverse the string is not dependent on the angle. I am sure that is the case, yet do not quite understand why.
My equations yield: L = 1/2*gcos(beta)*t^2, where L is the length of the string. Unless L were actually L*cos(beta), I really cannot see how t would not depend on the angle. But why would L be L*cos(beta), whereas the x-axis was defined to be parallel to the string? Would you please clarify?

 

[lewando]

Homework Statement

[/b]
...A smooth string is extended between two points A and B in a vertically positioned circle, so that the angle between the string and the vertical axis of the circle is denoted β. A bead of mass m slides from rest down a smooth chord in a vertically positioned circle...

This is a repost of a question whose thread was fairly badly mangled for multiple reasons. It makes sense to start fresh. One thing that should have been learned is to post all existing information about the problem. If you have a diagram that shows any constraints on the location of point A, please share. Otherwise it looks like points A and B are free to be anywhere on the circle.

My equations yield: L = 1/2*gcos(beta)*t^2, where L is the length of the string.

The length of the string is variable and is a function of the location of points A and B (or more directly, the length can be considered a function of angle β).

Unless L were actually L*cos(beta),

This might be what is confusing. Saying "L is Lcos(β)" makes little sense. Saying something like "L(β) = L_maxcos(β) would make better sense.

I really cannot see how t would not depend on the angle. But why would L be L*cos(beta), whereas the x-axis was defined to be parallel to the string? Would you please clarify?

Again, "L be L*cos(beta)", a confusing way of defining L.

 

[peripatein]

I couldn't agree more, and thank you for all your comments.
Please, would you kindly explain why the distance traversed along the x-axis (hence, along the string) is equal to L*cos(beta), where L denotes the length AB? Note that I chose the x-axis to be parallel to the string and thus to the motion of the bead along the string.

 

[lewando]

I couldn't agree more, and thank you for all your comments.
Please, would you kindly explain why the distance traversed along the x-axis (hence, along the string) is equal to L*cos(beta), where L denotes the length AB? Note that I chose the x-axis to be parallel to the string and thus to the motion of the bead along the string.

Let's look at what L*cos(β) means. If you have a chord AB with A at the "top" of the circle and B somewhere else, say located on the circle by angle β = CAB (with C at the center) then a right triangle can be formed, ADB, with D placed appropriately at a point along the line formed by AC. Call the height of this triangle "h" (the length of segment AD). Then cos(β) = adj/hyp = AD/AB = h/L. Then L*cos(β) simply equals h.

Trying to equate the phrase "the distance traversed along the x-axis (hence, along the string)" to "h" is something I can't do.

 

[peripatein]

Your example was very clear, however how is that helpful in demonstrating that the distance traversed should indeed be the length of the string multiplied by the cosine of the angle beta? It would indeed be the case if my x-axis were horizontal. But it isn't; it is, as you recall, parallel to the string and to the movement of the bead along the latter.
What would you suggest?

 

[peripatein]

It's a bit odd no one else has joined this discussion, helping to clarify this matter.

 

[lewando]

It's a bit odd no one else has joined this discussion, helping to clarify this matter.

Its the end of the year, a lot of people are on break, holiday social engagements, etc.

Your example was very clear, however how is that helpful in demonstrating that the distance traversed should indeed be the length of the string multiplied by the cosine of the angle beta? It would indeed be the case if my x-axis were horizontal. But it isn't; it is, as you recall, parallel to the string and to the movement of the bead along the latter.
What would you suggest?

Consider that the wording is not clear: "the distance traversed should be the length of the string multiplied by the cosine of the angle beta". Specifically unclear: is not the the "distance traversed" the same as the length of the string "L"? Maybe you mean "distance traversed" is something like x(t), a position along AB as a function of t. Except maybe you mean x(β). But here's the thing about that-- for a specific fixed chord, β is also fixed--t is the only independent variable. It is only when you are describing all chords, that β becomes an independent variable.

So, can you say it a different way, draw something, or just simply define your terms better? You are closer than you may think you are.

 

[peripatein]

I am not sure I follow. Would you agree that once the x-axis has been set to be parallel to the string, the distance covered along the x-axis by the bead sliding from A to B would simply be the length of the string, without any dependence on the angle?

 

[lewando]

I am not sure I follow. Would you agree that once the x-axis has been set to be parallel to the string, the distance covered along the x-axis by the bead sliding from A to B would simply be the length of the string, without any dependence on the angle?

Yes, for a specific chord, the chord has a length, L_{specific-chord}. It happens to be at an angle, β_{specific-chord}. You can even find t_{specific-chord}.

 

[peripatein]

But doesn't the acceleration along that same x-axis depend on beta? If it does, then how could t_{specific-chord} not depend on beta?

 

[lewando]

Yes acceleration depends on β. When β is 0°, acceleration is at a maximum and L is also at maximum. When β is 90°, acceleration is at a minimum, so is L. Time, supposedly, is independent of β. We are close to demonstrating that.

 

[peripatein]

What do you mean 'we'? Do you not already have the answer and are simply guiding me through? :-)

 

[lewando]

By "we" I mean "you" ;) One of the frustrating things about this site is that the answer can't just be blurted out. I understand why--you have pride is one reason. Another is to be a good teacher (or just to be good at helping you learn) I have to bring you to the place where truth lives, ring the doorbell and run away and let you answer the door.

 

[peripatein]

I simply cannot fathom how x=length of string=1/2*a*t^2 and a=g*cos(beta), lead to an expression for t which is not dependent on beta. What am I doing wrong?

 

[lewando]

I simply cannot fathom how x=length of string=1/2*a*t^2 and a=g*cos(beta), lead to an expression for t which is not dependent on beta. What am I doing wrong?

In order to demonstrate time independence for all β, you need to be as general as possible. Take "x=length of string=1/2*a*t^2" and "a=g*cos(beta)". You have generalized a, but not the length of the string (the left side of that equation). You need to be able to express the length of the string as a function of β (or cos(β), even better). To do that, use some trig, use the diameter (the maximum size of a chord), and draw a bunch of pictures until it clicks. I will supply a picture if I can find my Photobucket password.

 

[lewando]

Take a look at this--using a trig function relate β, L/2 and D/2 and see whet you come up with.

http://i1209.photobucket.com/albums/cc398/lewando/beta_zpsa3b56871.png

 

[rcgldr]

You need to be able to express the length of the string as a function of β (or cos(β), even better).

and r the radius of the circle. Assuming t is independent of β, then t is a function of g and r.