Particle experiencing only an angular force, determine the r dot

[flinnbella]

Homework Statement

Consider a particle that feels an angular force only, of the form Fθ = m r' θ'.
Determine the dependence of r' on r.

Relevant Equations

Relevant equations are:
particle acceleration in polar coordinates
Fr = 0
F(theta) = mr'θ'.

Hey, I've been working on this for a couple hours, and still no luck.

Since the force in the radial direction is zero, I set
r'' = rθ'^2.
Then since Fθ = m r' θ' and, since it's in polar coordinates, Fθ = m(2r'θ' + rθ'').
Setting these two equal, I get: -r'θ' = rθ''

At this point, I'm stumped. I try to substitute the angular velocity/ acceleration for something in terms of r, try to integrate, but inevitably I reach a point where I can't integrate anymore.

 

[PeroK]

Homework Statement: Consider a particle that feels an angular force only, of the form Fθ = m r' θ'.
Determine the dependence of r' on r.
Relevant Equations: Relevant equations are:
particle acceleration in polar coordinates
Fr = 0
F(theta) = mr'θ'.

Hey, I've been working on this for a couple hours, and still no luck.

Since the force in the radial direction is zero, I set
r'' = rθ'^2.
Then since Fθ = m r' θ' and, since it's in polar coordinates, Fθ = m(2r'θ' + rθ'').
Setting these two equal, I get: -r'θ' = rθ''

You can rewrite that as$$r\ddot \theta + \dot r \dot \theta = 0$$Do you recognise an exact time derivative there?

 

[PeroK]

PS you also have another equation of motion from the $\hat r$ component.

 

[PeroK]

PPS is the question to get the dependence of $\ddot r$ on $r$? I.e. a differential equation for $r$.

 

[erobz]

PPS is the question to get the dependence of $\ddot r$ on $r$? I.e. a differential equation for $r$.

As far as I can tell it's either what you suggest ( which is very clean ), or you get $\dot r$ as a function of $r, \ddot r , \dddot r$.

 

[Lnewqban]

What can be considered "an angular force" in this type of problems?

 

[kuruman]

A force that has an angular but no radial component?

 

[haruspex]

A force that has an angular but no radial component?

So $\vec F=m\dot r\dot\theta\hat\theta$, right @flinnbella ?

 

[flinnbella]

So $\vec F=m\dot r\dot\theta\hat\theta$, right @flinnbella ?

Yes exactly. There is no radial force and the radial acceleration is zero

 

[flinnbella]

PPS is the question to get the dependence of $\ddot r$ on $r$? I.e. a differential equation for $r$.

No its on the dependence of the radial velocity on the radial position.

 

[flinnbella]

PPS is the question to get the dependence of $\ddot r$ on $r$? I.e. a differential equation for $r$.

It's on r dot dependence on r, not r double dot

 

[PeroK]

It's on r dot dependence on r, not r double dot

It's difficult to know what is required, but if we take $\frac d {dt} \dot r = \ddot r$, then that gives us a relationship between $\dot r$ and $r$.

You should be able to make progress in any case, following the conventional approach in these problems (as I hinted at in the posts above).

 

[haruspex]

No its on the dependence of the radial velocity on the radial position.

You may recall $\ddot x=\dot x\frac{d\dot x}{dx}$.

 

[erobz]

You may recall $\ddot x=\dot x\frac{d\dot x}{dx}$.

Behold the power of the Chain Rule!

 

[flinnbella]

Behold the power of the Chain Rule!

Wow, I figured it out, thank you

 

[erobz]

Wow, I figured it out, thank you

Thanks! but I’ll forward that to the providers of the key insights @PeroK , @haruspex