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.
Homework 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.

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)

• Delta2
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TSny

Homework Helper
Gold Member
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.

Last edited:
• Delta2

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

Homework Helper
Gold Member
This part of your first post looks correct to me You just need to substitute for $\omega_0$ in terms of $\omega_1$.

Last edited:

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!

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

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