Integration of acceleration in polar coordinates

[tent]

Homework Statement

I give a particle a circular trajectory at 2 units of distance from the origin, and an angular velocity of 4 rad/s, both constant. Then I derive the velocity to obtain the radial acceleration, -32. I then say, "Given -32 radial acceleration, can I obtain the original position vector with the appropriate initial conditions?"

Relevant Equations

Acceleration, velocity, position components in polar coordinates

I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image.

My attempts are the following, I proceed using 3 "independent" methods just as you would solve a Cartesian coordinates kinematic problem, by integrating the acceleration.
1) From the radial and angular acceleration, a system of 2 diff eqs. Integrating by parts the angular one.

2) Then, by Cartesian subsitution.

3) Finally by integrating the radial unit vector.

Everything seems impossible to integrate and I'm out of ideas. The initial conditions are irrelevant, though I'm aware you could probably deduce that this is a circular motion from them, and hence solve this easily. The point is to generalize this simple example to an acceleration of an arbitrary trajectory with both components where you shouldn't be able to heuristically deduce the velocity or position. However, I can't even solve this seemingly trivial case.

 

[PeroK]

Summary: Given acceleration in polar coordinates, find the position vector in polar. Integration problem.

It's not clear to me what you are trying to do. Could you pick one of your examples and explain what's going on?

 

[PeroK]

PS something like $\vec r = \rho(t) \hat r$ is not a well-defined trajectory. Can you see why?

 

[tent]

It's not clear to me what you are trying to do. Could you pick one of your examples and explain what's going on?

I give a particle a circular trajectory at 2 units of distance from the origin, and an angular velocity of 4 rad/s, both constant. Then I derive the velocity to obtain the radial acceleration, -32. I then say, "Given -32 radial acceleration, can I obtain the original position vector with the appropriate initial conditions?" And so I proceed to solve it using 3 methods just as you would solve a cartesian coordinates kinematic problem, by integrating the acceleration.

PS something like $\vec r = \rho(t) \hat r$ is not a well-defined trajectory. Can you see why?

The angle is missing, I assume, but it should be implicit in the radial unit vector, which depends on θ which is a function of t. So the way I understand it, it should be well-defined.

 

[PeroK]

I give a particle a circular trajectory at 2 units of distance from the origin, and an angular velocity of 4 rad/s, both constant. Then I derive the velocity to obtain the radial acceleration, -32.

$-32$ is just a number. In any case, the have $a_r = \ddot r - r\dot \theta^2 = k$ and $a_{\theta} = r\ddot \theta + 2\dot r \dot \theta = 0$.

Does that define a unique trajectory? Or, are there other trajectories that produce the same acceleration in polar coordinates?

I then say, "Given -32 radial acceleration, can I obtain the original position vector with the appropriate initial conditions?" And so I proceed to solve it using 3 methods just as you would solve a cartesian coordinates kinematic problem, by integrating the acceleration.

Try converting to Cartesian coordinates and see what happens.

The angle is missing, I assume, but it should be implicit in the radial unit vector, which depends on θ which is a function of t. So the way I understand it, it should be well-defined.

Can you draw a diagram showing the trajectory?

 

[tent]

Thread was moved over to https://www.physicsforums.com/threads/integration-of-acceleration-in-polar-coordinates.1046427/

 

[PeroK]

Thread was moved over to https://www.physicsforums.com/threads/integration-of-acceleration-in-polar-coordinates.1046427/

Given I've responded here, please continue with this thread. Thanks.

 

[PeroK]

PS if I give you the acceleration of a particle in Cartesian coordinates as$$\ddot x = a_x, \ddot y = a_y$$can you construct the trajectory by integration? Where $a_x, a_y$ are constant.

 

[pasmith]

In classical mechanics, circular motion with constant angular velocity $\omega$ satisfies $v_r = 0$ and $v_\theta = r\omega$. The acceleration is given by $$\frac{d}{dt}(v_\theta\mathbf{e}_\theta) = \dot{v_\theta}\mathbf{e}_\theta - v_\theta \dot \theta \mathbf{e}_r = -r\omega^2 \mathbf{e}_r.$$ Thus you want to solve $$\begin{split} \ddot r - r\dot\theta^2 &= - r \omega^2 \\ \frac{1}{r} \frac{d}{dt}(r^2 \dot \theta) &= 0.\end{split}$$ But we are assuming $r$ is constant, so the first equation reduces to $$\dot \theta^2 = \omega^2$$ and the second reduces to $$\ddot \theta = 0.$$ If you want to start from $$\begin{split} \ddot r - r\dot \theta^2 &= -k\\ \frac{d}{dt}(r^2 \dot \theta) &= 0 \end{split}$$ then the substitution $u = 1/r$ with $\dot r = -Lu'(\theta)$ where $L = r^2 \dot \theta$ is constant yields $$u'' + u = \frac{-k}{L^2u^2}$$ which admits a constant solution if $u = -(k/L^2)^{1/3} < 0$, so that is not the equation you are looking for. You can obtain circular motion from an inverse square central force with the right choice of constant of proportionality given the initial conditions.

 

[tent]

$-32$ is just a number. In any case, the have $a_r = \ddot r - r\dot \theta^2 = k$ and $a_{\theta} = r\ddot \theta + 2\dot r \dot \theta = 0$.

Does that define a unique trajectory? Or, are there other trajectories that produce the same acceleration in polar coordinates?

Try converting to Cartesian coordinates and see what happens.

Can you draw a diagram showing the trajectory?

I assume there are other trajectories, yes, the set of all values of $\ddot r, r,\dot \theta^2$ that give k.

PS if I give you the acceleration of a particle in Cartesian coordinates as$$\ddot x = a_x, \ddot y = a_y$$can you construct the trajectory by integration? Where $a_x, a_y$ are constant.

If this is what you're referring to, then yes I can see I need the initial conditions.Ok, so I did the position to acceleration for circular motion in Cartesian. Then I tried doing the reverse problem, expressed in Cartesian. I think I get it now, I neglected the initial conditions thinking I could solve the problem without them but it is impossible.

However adding these conditions in particular, that I started out with, makes the problem trivial.
So I decided to propose a new problem.

This is the expression of a general acceleration with constant components in polar coordinates. The initial conditions define the position of the particle at time 0 and its velocity components in polar coordinates. I believe these conditions specify a unique trajectory and should be enough to give a solution, a vector of the position, in polar coords. Looking at it, it is probably a spiral, but prior knowledge of the shape shouldn't be used in the solution as the intent is to be able to solve more complex expressions where you don't know the shape of the trajectory.

I attempted the following solution, but it led me nowhere.

This seems extremely complex, and transforming to Cartesian doesn't help as you don't know what θ is as a function of t, hence being impossible to integrate. Do I need to just commit and solve this mess, or am I not seeing something?

All I'm looking to do is find the position vector from integration of a given acceleration, just as you do in Cartesian, but doing it in polar coordinates somehow makes everything 100 times harder.

 

[PeroK]

You have $$\frac d{dt} \vec v = k_1 \hat r + k_2 \hat \theta$$Can you express that in Cartesian coordinates and then integrate?

I suspect you may have to transform to Cartesian coordinates to make progress.

 

[tent]

You have $$\frac d{dt} \vec v = k_1 \hat r + k_2 \hat \theta$$Can you express that in Cartesian coordinates and then integrate?

I suspect you may have to transform to Cartesian coordinates to make progress.

Okay, I tried integrating in polar coordinates and in Cartesian, but nothing seems to work.

I feel like I'm missing some condition. This can't be so hard to solve.

 

[PeroK]

I don't immediately see any way to make progress. It feels like you need a clever substitution.

 

[tent]

I don't immediately see any way to make progress. It feels like you need a clever substitution.

Is this how one proceeds, though? Do you reckon this is even solvable the way I'm trying to?

 

[PeroK]

Is this how one proceeds, though? Do you reckon this is even solvable the way I'm trying to?

I don't know. Perhaps not solvable in general.

 

[PeterDonis]

@tent, posting equations in images is not acceptable. You will need to use the PF LaTeX feature to post equations directly, as is done in the responses in this thread.

 

[tent]

@tent, posting equations in images is not acceptable. You will need to use the PF LaTeX feature to post equations directly, as is done in the responses in this thread.

Oh, sure. Apologies

 

[Vanadium 50]

There is a lot of confusion in this thread, and I think it comes from the problem statement:

"I give a particle a circular trajectory at 2 units of distance from the origin, and an angular velocity of 4 rad/s, both constant. Then I derive the velocity to obtain the radial acceleration, -32."

If it is at constant distance from the origin, the radial acceleration is zero, not -32. So the problem, as stated, is internally contradictory.

Maybe we can sleuth out what you intend, but at this point, I believe this thread is too much of a mess to be helpful to anyone. My advice would be:

• Have the mentors close this thead, and start again with a clearer question.
• Use TeX.
• Use equations, e.g. "r = 2 (constant)" or "r(t = 0) = 2". That will help us figure out what is meant.

 

[strangerep]

There is a lot of confusion in this thread, and I think it comes from the problem statement:

"I give a particle a circular trajectory at 2 units of distance from the origin, and an angular velocity of 4 rad/s, both constant. Then I derive the velocity to obtain the radial acceleration, -32."

If it is at constant distance from the origin, the radial acceleration is zero, not -32. [...]

Worded that way, I believe your last statement is not correct. The general formula for the radial component of the acceleration vector in spherical polar coordinates is: $$a_r ~=~ \ddot r - r \dot\theta^2 - r \dot\phi^2 \sin^2\theta ~. $$ For motion in a plane, we can keep $\phi$ constant, so this reduces to $$a_r ~=~ \ddot r - r \dot\theta^2 ~. $$ If $r$ is constant then $a_r = -r \dot\theta^2$. If the angular velocity $\dot\theta$ is constant at 4 rad/s, and the radius is 2 metres (say), then $a_r = - 2 \times 4^2 = -32 \, m/s^2$.

(If I understand the OP's intent correctly, this problem should be almost trivial. The most important thing is to begin by actually writing out the correct components of the acceleration vector in spherical polar coordinates -- which he/she seems not to have done, at least not clearly.)

 

[PeterDonis]

I give a particle a circular trajectory at 2 units of distance from the origin, and an angular velocity of 4 rad/s, both constant. Then I derive the velocity to obtain the radial acceleration, -32. I then say, "Given -32 radial acceleration, can I obtain the original position vector with the appropriate initial conditions?"

There doesn't seem to me to be much point to this since you have already specified the solution: your problem statement already gives you the first integral of acceleration (angular velocity constant at 4 rad/s) and the second integral except for a constant of integration (position at 2 units from the origin, i.e., $r = 2$; the only thing left is to pick the angular coordinates at $t = 0$, but since that's an integration constant it can be chosen arbitrarily and there is no information about it anywhere else). So you have no need to solve anything.

 

[PeterDonis]

If it is at constant distance from the origin, the radial acceleration is zero, not -32.

No, the radial acceleration is indeed - 32, as @strangerep has said. Circular motion means a constant radius from the origin but not a zero radial acceleration; it just means the acceleration vector is perpendicular to the velocity vector.

There is a lot of confusion in this thread, and I think it comes from the problem statement

I think the problem statement, or at least the one you quoted, is indeed pointless, but not for the reason you give. See my post #20 just now.

 

[pasmith]

You have $$\frac d{dt} \vec v = k_1 \hat r + k_2 \hat \theta$$Can you express that in Cartesian coordinates and then integrate?

I suspect you may have to transform to Cartesian coordinates to make progress.

I think you end up with $$\begin{split} \ddot r &= k_1 + L^2r^{-3} \\ \dot L &= k_2 r \end{split}$$ where $L =r^2 \dot \theta$ subject to $$r(0) = a\qquad\dot r(0) = 0 \qquad L(0) = a^2\omega$$ which is not solvable analytically. In the $k_2 = 0$ (constant $L = a^2\omega$) case you can get a first integral $$\dot r^2 = \frac{4k_1r^5 + Cr^4 - a^4\omega^2}{2r^4}$$ which is again not solvable analytically in general, but with our initial conditions we have $$\dot r^2 = \frac{4k_1r^4(r - a) + \omega^2(r^4 - a^4)}{2r^4}.$$
At this point you can conclude that $r(t) \equiv a$ and go on to find $\dot \theta(t) \equiv \omega$.

 

[Vanadium 50]

While I think we're smart enough to figure out what the OP meant, I still think making him write a clearer question is in everybody's interest. He will get more and better help, he will get it faster, and he will learn to write good questions.

 

[tent]

My intent with this question was to see how you integrate acceleration in polar coordinates. Whether it's as easy as in Cartesian coordinates because it constantly changes orientation with the polar unit vectors which adds difficulty. To do that, I defined an acceleration for which I already know the position vector, by differentiating from the position vector so that I can check my answer in the end. The idea is to work backwards from that acceleration vector to get to the position vector.
Now, depending on the initial conditions, this could be trivial (by giving the angular speed, it's easy to integrate back to the position vector) or it could be seemingly impossible (by giving the initial position and velocity components, the equations become too complex for me, though I believe it's a well defined trajectory with a unique solution). The question is whether it is possible to solve analytically for an acceleration with constant polar components and initial position + initial velocity components.

 

[PeterDonis]

@tent now that you have explained what you are trying to do, two things are clear:

(1) This is not a homework problem and does not belong in the homework forum. It's a classical physics problem and belongs in the classical physics forum.

(2) You need to start a new thread in the correct forum and post your equations directly in the thread using the PF LaTeX support instead of as images.You also need to pick one specific scenario.

Given the above, this thread is closed.