Caculating velocity on a different points on a rim of a wheel- AGAIN

[mmoadi]

Homework Statement

Motorcycle is accelerating from the rest with a constant acceleration a = 2 m/s². We are observing its front wheel (see picture) after a time t = 10 s. This wheel is not slipping while accelerating. How much are velocities of points A, B, and C on rim of wheel? How much are the accelerations of these points?

My first attempt of solving the problem:

v= v0 + at

A: v = v0 + at → v= -20 m/s
B: v= sqrt(vA² + vB²) → 28.3 m/s
C: v= v0 + at → v= 20 m/s

This is the question I submitted yesterday. And after getting some useful hints:

Quotes:
Doc Al said: "Hint: Find the velocity of the center of the wheel with respect to the ground, the velocity of the rim with respect to the center, then the velocity of each part of the rim with respect to the ground."
Thank you!

and

turin said: "There is a rotational part that you seem to be neglecting, and when you include it, you may be quite surprised, especially about point C. Also, don't forget that the problem asks for the accelerations, too."
Thank you!

I did some research and this is what I found out:

1. A wheel rolling over a surface has both a linear and a rotational velocity.
2. The linear velocity of any point on the rim of the wheel is given by vcm= ωr.
3. Every point on the rotating object has the same angular speed.
4. Because when the wheel is in contact with the ground, its bottom part is at rest with respect to the ground, the wheel experiences a linear motion with a velocity equal to + vcm besides a rotational motion (picture).
5. Conclusion: the top of the wheel moves twice as fast as the center and the bottom of the wheel does not move at all.

Relevant equations:

Angular speed:ω= Θ / t
Angular acceleration:α= ω / t
Tangential speed: vt= rω
Tangential acceleration: at= rα

The only problem now is:How to calculate radius r?

I would really like to solve this problem. Thank you for helping!

 

[mmoadi]

OK, I think I got it.
It's a linear acceleration problem so I will simply put the rotational formulas aside.

v= v_i + at where v_i = 0
So, v= 2 m/s² * 10 s= 20 m/s

For point A: v_gr= 2v= 40 m/s → a= 4 m/s²
For point B: v_gr= √2 v= 28.28 m/s → a= 2.83 m/s²
For point C: v_gr= 0 m/s → a= 0 m/s²

And I really really hope it's a correct answer!

 

[tiny-tim]

… The only problem now is:How to calculate radius r?

Hi mmoadi!

You don't need to know r.

Just leave r as r … you'll find it disappears at the end (except for αr, which you know is 2 m/s²).

 

[mmoadi]

Can you take a look at my above calculations? Are they correct?

 

[tiny-tim]

Can you take a look at my above calculations? Are they correct?

(ooh, i didn't see them …)

[STRIKE]A and C are correct (though you could have got them just from (a_A + a_C)/2 = a_com )[/STRIKE]

For B …

For point B: v_gr= √2 v= 28.28 m/s → a= 2.83 m/s²

… how did you get that (v is right, but I don't think a is)?

EDIT: Changed my mind … see below.​

 

[mmoadi]

I just calculated a= v_gr / t = (28.28 m/s) / 10 s= 2.83 m/s²

 

[mmoadi]

This is how I concluded how to calculate the velocity in respect to the ground.

 

[tiny-tim]

A and C are correct (though you could have got them just from (a_A + a_C)/2 = a_com )

(i can't see your new picture yet)

On second thoughts, I don't think those accelerations are right, after all (though the velocities certainly are) …

they don't take the centripetal acceleration into account, and I don't see how you can find that without knowing r (as you originally mentioned)

 

[mmoadi]

So, any suggestion how can I find R?

 

[tiny-tim]

So, any suggestion how can I find R?

No … if the question doesn't give you R, you can solve for the velocities, but not I think for the accelerations.

 

[mmoadi]

So, there is no way to calculate r from the quantities given?

 

[tiny-tim]

I can't see any.

 

[mmoadi]

Thank you very much for all the patience and help!

Is it possible to imply for r just an average radius for a motorcycle tire and solve the problem to the end?