• Support PF! Buy your school textbooks, materials and every day products via PF Here!

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

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.
Skärmavbild 2019-10-10 kl. 00.45.01.png

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:


IMG_0419.jpg

IMG_0420.jpg


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

TSny

Homework Helper
Gold Member
12,037
2,619
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:
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:

IMG_0421.JPG


IMG_0422.JPG


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
12,037
2,619
This part of your first post looks correct to me
1570817256729.png


You just need to substitute for ##\omega_0## in terms of ##\omega_1##.
 
Last edited:

Want to reply to this thread?

"Write an expression w1 of the angle shown in the first picture" You must log in or register to reply here.

Related Threads for: Write an expression w1 of the angle shown in the first picture

Replies
12
Views
2K
  • Posted
Replies
7
Views
1K
Replies
10
Views
687
Replies
4
Views
328
Replies
1
Views
429
Replies
5
Views
17K
Replies
8
Views
772

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top