Task: Function for the acceleration throughout a loop?

[OskarBillington]

So, this seemed really fun to me until I got stuck.

THE TASK is about an object with mass m, moving in a basic (2D) coordinate system. It is attached to origo (0, 0) by a "rope" with constant length r=5. In position P₀(-5, 0) it has the velocity v₀=[0, -10]. Hence, the object moves around origo in a circular loop, and the force on m from the rope shall be referred to as C. There is also another (constant) force: F=[10m, 0] (m being the mass: all calculations should be done without units). F could represent a gravitational pull, to ease the comparison to a standard loop. By the terms, the mechanical energy is conserved.

First, you are asked to find the acceleration (expressed as a vector) in points (-5, 0), (0, -5), (5, 0), (0, 5). This is easily done by Newton's 2nd law: ΣF=ma⇒ either m[±v²/r, 0] (in points (-5, 0) and (5, 0), because ΣF=centripetal force=mv₂/2) or m[10, ±v²/r] (in the other two points, because ΣF_x=F and ΣF_y=centripetal force).

FINALLY: You are now asked to find a function a(α)=[a_x, a_y].
- a is the acceleration vector of m.
- α is the angle, determining the position of m. Preferably, you want this to be the angle from the positive x-axis to the "rope" (counter clockwise).
- The vector function shall describe the acceleration vector of m at any position in the loop.
Good luck

Notes:

It seems that the reason why the second task is hard is because F interferes with C_x. In comparison: in the first task, F has either everything or nothing to do with C. C does of course consist of the pull from the rope, and in most positions also some push/pull from F. I can hardly explain the issue any better (at least in English), but you may anyhow find that the task is quite difficult.

An idea of mine is to define the x-/ y-axes along the velocity vector and the C vector, so that when α changes, F only rotates around m. In this redefined coordinate system: ΣF_x will then depend only on the angle, and will simply* be equal to F_x (*still difficult to cover the entire 360° movement with one function). ΣF_y will always equal C, again equal to mv²/r - and a function of the size v may be possible to work out. With this x-/y-perspective I suppose you will result in a vector function for a, which would then have to be modified to "rotate" back to work with the x-/y-axis definition in the original task.

I am inexperienced with the English physics terms, hence the poor or incorrect use of terminology. Please inform me on potential for improvement!

 

[Drakkith]

Hi Oskar,

I've moved this thread to the introductory physics homework forum, but in the future please post all homework or homework-type questions in the appropriate homework forum and use the template provided when making a new thread. Thanks.

 

[haruspex]

As the question suggests, this is just like a mass executing a vertical circle in the presence of gravity. How would you determine the speed at a given angle in that case?

 

[OskarBillington]

As the question suggests, this is just like a mass executing a vertical circle in the presence of gravity. How would you determine the speed at a given angle in that case?

At a given angle, I don't know. At the 4 angles in task 1 I thought I was onto something (W_F=Fs=Δmv²/2). However, I just realized that, i.e., F does not add 10m*10 kinetic energy units from P₀ to (5, 0), as F does work on both the speed and C all the way. Working on general velocity now...

 

[haruspex]

At a given angle, I don't know. At the 4 angles in task 1 I thought I was onto something (W_F=Fs=Δmv²/2). However, I just realized that, i.e., F does not add 10m*10 kinetic energy units from P₀ to (5, 0), as F does work on both the speed and C all the way. Working on general velocity now...

Will the string remain taut? If work is done against C, what does that imply about the string length?