Motion of a point mass (circular motion)

[Carson Birth]

Homework Statement

A point mass, sliding over an even, horizontal plane is bounded on an inextensible, massless thread. During the motion, the thread is pulled by a force F with constant velocity $v_{o}$ through a hole O. In the beginning ($t_{o}$ = 0) r($t_{o}$)=b holds (the length of the thread on the plane is b at the beginning then will get shorter due to the force F as it is pulled through). The initial velocity of the point mass perpendicular to the thread is $v_{1}$ in $\phi$ direction, and the angle $\phi$($t_{o}$) = $\phi_{o}$ = 0Formulate the system's equation of motion and the equation of constraint forces in polar coordinates. Apply Newton's Law. Also the polar coordinate system is attached to the masspoint with r pointing away from the hole and $\phi$ pointing toward the trajectory.

Homework Equations

$\overrightarrow{a}$ = ($\ddot{r}$ - r$\dot{\phi}^2$) in r direction + (r$\ddot{\phi}$+2$\dot{r}$$\dot{\phi}$) in $\phi$ direction

The Attempt at a Solution

I separated the forces into there respected directions:
$\ddot{\phi}$ direction: mr$\ddot{\phi}$ + m2$\dot{r}$$\dot{\phi}$ = 0
r direction: F + mr$\dot{\phi}^2$ = 0

Now I am pretty sure there isn't any constraint forces, since there is no N force effecting the mass point.

So now I need to create an equation of motion, and from my understanding I need to create one equation. Is it as simple as just solving for $\dot{\phi}$ and plugging it into the other equation? I had a similar equation where I solved the $\phi$ direction equation as a differential but it didnt have the 2$\dot{r}$$\dot{\phi}$ term with it, it was a gravity force making it very simple to solve. I know this isn't for people to solve my homework so I am just looking for advice on how to get it all set up for the further sub-questions. Any advice would be appreciated :D

 

[ehild]

Homework Statement

A point mass, sliding over an even, horizontal plane is bounded on an inextensible, massless thread. During the motion, the thread is pulled by a force F with constant velocity $v_{o}$ through a hole O. In the beginning ($t_{o}$ = 0) r($t_{o}$)=b holds (the length of the thread on the plane is b at the beginning then will get shorter due to the force F as it is pulled through). The initial velocity of the point mass perpendicular to the thread is $v_{1}$ in $\phi$ direction, and the angle $\phi$($t_{o}$) = $\phi_{o}$ = 0Formulate the system's equation of motion and the equation of constraint forces in polar coordinates. Apply Newton's Law. Also the polar coordinate system is attached to the masspoint with r pointing away from the hole and $\phi$ pointing toward the trajectory.

Homework Equations

$\overrightarrow{a}$ = ($\ddot{r}$ - r$\dot{\phi}^2$) in r direction + (r$\ddot{\phi}$+2$\dot{r}$$\dot{\phi}$) in $\phi$ direction

The Attempt at a Solution

I separated the forces into there respected directions:
$\ddot{\phi}$ direction: mr$\ddot{\phi}$ + m2$\dot{r}$$\dot{\phi}$ = 0
r direction: F + mr$\dot{\phi}^2$ = 0

r is changing. Why did you ignored $\ddot r$?

 

[Carson Birth]

r is changing. Why did you ignored $\ddot r$?

My thought was since the thread is being pulled by a force with constant velocity, that it wouldn't be accelerating.

 

[ehild]

My thought was since the thread is being pulled by a force with constant velocity, that it wouldn't be accelerating.

You are right, I misread it as "constant force". Sorry.
So you do not know F, but you know that $\dot r$ is constant. Go ahead. Solve the first equation.

 

[Carson Birth]

You are right, I misread it as "constant force". Sorry.
So you do not know F, but you know that $\dot r$ is constant. Go ahead. Solve the first equation.

When you say solve the first equation, what do you mean? Would I solve the first equation as a differential equation or you mean solve for a variable then put it into the first equation? :D

 

[ehild]

When you say solve the first equation, what do you mean? Would I solve the first equation as a differential equation or you mean solve for a variable then put it into the first equation? :D

It is a differential equation for Φ as function of time. You know $\dot r$ hence also r(t).