Problem about non-inertial reference frame

[MatinSAR]

Homework Statement

Consider two circles(in a plane) rotating with a constant angular velocity, ω, relative to an inertial reference frame. The center of one circle is fixed at the origin of this reference frame, while the center of the second circle is located on the circumference of the first circle. If there is a point mass located on the second circle, what would be the fictitious force acting on it? The position vector of the point mass is $\vec R(t)$ in C1 circle.

Relevant Equations

Newton's 2nd law in non-inertial reference frame.

Picture of the problem:

I wanted to use this formula: (From Classical dynamics of particles and systems Book by Stephen Thornton)

O is inertial observer. O' is non-inertial observer.
I think $F$ and $\ddot R$ and $\dot \omega$ are $0$. According to O' the point mass has velocity of $v_r$ so it's not $0$. I believe I just need to determine the values of $w \times r$ and $\omega \times v_r$.

However, I’m uncertain about how to proceed. I would appreciate any assistance you could provide.

 

[haruspex]

Your choice of non-inertial frame is not entirely clear. Is it:

• origin O' orbits O but the axes are in fixed directions,
• origin O' orbits O and axes rotate anticlockwise at rate $\omega$, or
• origin remains at O but axes rotate anticlockwise at rate $\omega$

?

 

[MatinSAR]

• origin O' orbits O but the axes are in fixed directions

This is what I meant.

 

[haruspex]

I would answer the question by comparing the accelerations in the two reference frames. The fictitious force would be held responsible for the difference.

 

[MatinSAR]

The fictitious force would be held responsible for the difference.

Very meaningful. I didn't look at it this way. I will try to solve again.

 

[MatinSAR]

I would answer the question by comparing the accelerations in the two reference frames. The fictitious force would be held responsible for the difference.

Can I say that the point mass rotates with angular velocity of $\vec \omega$ compare to O' and $2 \vec \omega$ compare to O?

 

[haruspex]

Can I say that the point mass rotates with angular velocity of $\vec \omega$ compare to O' and $2 \vec \omega$ compare to O?
View attachment 345791

No, you can’t add rotations about different centres. The motion of the point about O varies over time. If the radii are $r_1, r_2$ then the motion varies between
- a tangential speed of $\omega r_2+\omega(r_1+r_2)$ at distance $r_1+r_2$, implying an angular rotation of $\omega \frac{ r_1+2r_2}{r_1+r_2}$ about O, and
- a tangential speed of $-\omega r_2+\omega(r_1-r_2)$ at distance $r_1-r_2$, implying an angular rotation of $\omega \frac{ r_1-2r_2}{r_1-r_2}$ about O.

 

[MatinSAR]

@haruspex I'm going to solve this question using TA's help. I will share the results. Thank you for your help and time.

 

[haruspex]

@haruspex I'm going to solve this question using TA's help. I will share the results. Thank you for your help and time.

Ok, but if you write down the vector expression for the position of the mass at time t you can differentiate to find the acceleration. You can do that in both the ground frame and the accelerating frame.