Dynamics of a point mass in circular motion

[Krzysiek Sam]

Homework Statement

Dear All,

I'm having a hard time solving the following problem:

A point of mass is moving on a circular plane (Oxy), where the circle's formula is:

The force acting on mass "m" is defined as:

We're looking for velocity of point "m" in position (1,1) =V1, and in position (0,2)=V2, given the velocity in position (0,0)=V0.

Homework Equations

(as above)

The Attempt at a Solution

m*x''=-k*x*y²
m*y''=-k*y*x²

x = r*cos(φ)
y=1+r*sin(φ)

φ = ω*t
x'= -r*ω*sin(ωt)
x''= -r*ω²*cos(ωt)

y'= r*ω*cos(ωt)
y''= -r*ω²*sin(ωt)

This leads me to a second order differential equation which I'm not able to solve.

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Thank you in advance for any help!

 

[TSny]

Hello, and welcome to PF!

φ = ω*t

Why would $\phi$ increase proportionally to time?

Another approach: Are you familiar with the concept of a "conservative force" and the potential energy associated with a conservative force?

EDIT: I'm not completely clear on the setup of the problem. Is the particle confined to move on the circle? Can we think of it as a bead sliding on a circular wire with no friction, but with an applied force as given?

 

[Krzysiek Sam]

Hello, and welcome to PF!
Why would $\phi$ increase proportionally to time?

Another approach: Are you familiar with the concept of a "conservative force" and the potential energy associated with a conservative force?

EDIT: I'm not completely clear on the setup of the problem. Is the particle confined to move on the circle? Can we think of it as a bead sliding on a circular wire with no friction, but with an applied force as given?

Hello,
1. phi increases, as m moves around the circle, phi[rad] = omega[rad/s]*t

2. how would you approach this with respect to conservative force?

3. yes, it's confined to move on the edge of the circle with radius r=1, yes the problem considers no friction, only forces Fx and Fy.

Thanks in advance for your help.
KS

 

[TSny]

1. phi increases, as m moves around the circle, phi[rad] = omega[rad/s]*t

Yes, $\phi$ increases as the mass moves around the circle. But, $\phi$ doesn't necessarily increase at a constant rate. So, you cannot assume $\phi = \omega t$ for some constant $\omega$.

2. how would you approach this with respect to conservative force?

First, I would check to see if the given force is conservative. Have you learned how to do that?

 

[Krzysiek Sam]

Yes, $\phi$ increases as the mass moves around the circle. But, $\phi$ doesn't necessarily increase at a constant rate. So, you cannot assume $\phi = \omega t$ for some constant $\omega$.

First, I would check to see if the given force is conservative. Have you learned how to do that?

Yes, this is the right approach, thank you for your suggestion!
Have a nice day!