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- 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.

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)