Write an expression w1 of the angle shown in the first picture

[Karl Karlsson]

Homework Statement

A bicycle wheel rolls at a constant speed along a circular path on a horizontal surface. The wheel has a constant angle of inclination to the vertical direction and the distance from its center of mass G to the fixed Z axis is R. Determine the relationship between the angular velocity w1 around the Z axis and the angle of inclination (shown in the picture above). Treat the wheel as a homogeneous ring with mass m and radius r.

Relevant Equations

A bicycle wheel rolls at a constant speed along a circular path on a horizontal surface. The wheel has a constant angle of inclination to the vertical direction and the distance from its center of mass G to the fixed Z axis is R. Determine the relationship between the angular velocity w1 around the Z axis and the angle of inclination (shown in the picture above). Treat the wheel as a homogeneous ring with mass m and radius r.

A bicycle wheel rolls at a constant speed along a circular path on a horizontal surface. The wheel has a constant angle of inclination to the vertical direction and the distance from its center of mass G to the fixed Z axis is R. Determine the relationship between the angular velocity w1 around the Z axis and the angle of inclination (shown in the picture above). Treat the wheel as a homogeneous ring with mass m and radius r.

Lead:
1) Introduce a resale system Gxyz as shown in the figure.
2) Use the kinematics (speed relationship between G and C) and determine the relationship between
wheel spinning speed ω0 around x-axis and ω1. Consider the direction of ω0.
3) Formulate the force equation maG F and determine the frictional force F and the normal force N of
the wheel at the contact point C.
4) Determine the wheel's torque HG = IGω in the resal system. What is ω here?
5) Formulate the torque equation HG  ωS  HG  MG. What is ωS here?
6) Insert the relationship between 0 and 1 in the torque equation and determine 1.

My attempt:

The correct answer is w1=(2gtan(angle)/(4R+r*sin(angle)))^(1/2)

 

[TSny]

I think the mistake is with the relation $\omega_0 = \frac{R}{r} \omega_1$.

I believe the correct relation is $\omega_0 = \frac{R+r\sin\theta}{r} \omega_1$.

In your derivation where you have an expression for $\vec V_c$, I think you should have

$\vec V_c =\omega_x \hat e_x \times (-r \hat e_z)+R \omega_1 \hat e_y = (-\omega_0 +\omega_1\sin \theta)\hat e_x \times (-r \hat e_z)+R \omega_1 \hat e_y$.

Here, $\omega_x$ is the x-component of the vector $\vec \omega$ that you used in the matrix calculation of the angular momentum. From $\vec V_c = 0$, you then get $\omega_0 = \frac{R+r\sin\theta}{r} \omega_1$.

If you use this expression for $\omega_0$ in the next-to-last line of your first page of notes, then I think things will work out.

If possible, we ask that you please try to type out your work rather than post a picture of your hand written notes. It is difficult to quote a specific part of a picture.

 

[Karl Karlsson]

I think the mistake is with the relation $\omega_0 = \frac{R}{r} \omega_1$.

I believe the correct relation is $\omega_0 = \frac{R+r\sin\theta}{r} \omega_1$.

In your derivation where you have an expression for $\vec V_c$, I think you should have

$\vec V_c =\omega_x \hat e_x \times (-r \hat e_z)+R \omega_1 \hat e_y = (-\omega_0 +\omega_1\sin \theta)\hat e_x \times (-r \hat e_z)+R \omega_1 \hat e_y$.

Here, $\omega_x$ is the x-component of the vector $\vec \omega$ that you used in the matric calculation of the angular momentum. From $\vec V_c = 0$, you then get $\omega_0 = \frac{R+r\sin\theta}{r} \omega_1$.

If you use this expression for $\omega_0$ in the next-to-last line of your first page of notes, then I think things will work out.

If possible, we ask that you please try to type out your work rather than post a picture of your hand written notes. It is difficult to quote a specific part of a picture.

Thanks! Of course, I must have forgotten about that. But now, for some reason, I still don't manage to get the correct answer:

There are so many equations and a matrix involved which makes it a lot easier for me to send a picture compared to writing all the equations on the computer

 

[TSny]

This part of your first post looks correct to me

You just need to substitute for $\omega_0$ in terms of $\omega_1$.

 

[Karl Karlsson]

This part of your first post looks correct to me
View attachment 250981

You just need to substitute for $\omega_0$ in terms of $\omega_1$.

That's right, it works now. Thanks!