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

• Karl Karlsson
In summary, the conversation discusses a bicycle wheel rolling at a constant speed on a horizontal surface with a fixed Z-axis and a constant angle of inclination. The relationship between the wheel's angular velocity w1 around the Z-axis and the angle of inclination is determined by treating the wheel as a homogeneous ring with mass m and radius r. The conversation also discusses the force and torque equations for the wheel, as well as the correct relationship between the wheel's spinning speed w0 around the x-axis and w1. The correct relationship is found to be w0 = (R+r*sin(angle)/r)w1. Finally, the conversation touches on the use of equations and matrices in finding the correct solution.
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.

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:

Delta2
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
TSny said:
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

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:
TSny said:
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!

## 1. What is an expression for the angle shown in the first picture?

The expression for the angle shown in the first picture is represented by w1.

## 2. How do you write an expression for an angle?

To write an expression for an angle, you can use the variable w followed by a number to represent the angle. For example, w1 would represent the first angle, w2 would represent the second angle, and so on.

## 3. Can you explain the significance of w in the expression for the angle?

The variable w is used to represent an unknown angle in the expression. It allows us to generalize the expression and use it for any angle in a given situation.

## 4. How is the angle shown in the first picture related to the other angles in the problem?

The angle shown in the first picture is related to the other angles in the problem as it is a part of a larger geometric figure or shape. Its measurement and relationship to the other angles can be determined by using mathematical principles and formulas.

## 5. Is there a specific unit of measurement for the angle in the expression?

The angle shown in the first picture can be measured in degrees, radians, or other units depending on the context of the problem. The expression w1 does not specify a specific unit of measurement and can be used for any unit of measurement for angles.

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