Velocity from position vector in rotating object

In summary, the conversation discusses solving for the change in velocity for the center of a rim with respect to the contact patch of a tire with camber. The equation 2.6 is derived from 2.5, with the first part being the derivation of 2.5 with respect to h. The second and fourth terms in 2.6 involve the change in j and k with respect to time, which are related to the yaw moment and the tangent of gamma respectively. The third term uses the i unit vector for mathematical reasoning.
  • #1
Nikstykal
31
1
Q1.PNG

Homework Statement


I am trying to solve for change in velocity for the center of a rim with respect to the contact patch of a tire that has some degree of camber. The equation finalized is shown in the image below, equation 2.6.
http://imgur.com/a/oHucp

Homework Equations

The Attempt at a Solution


I understand how to get the position vector shown in 2.5. The first part of 2.6 is just deriving 2.5 with respect to h. The 3rd and 4th terms are what confuse me. In regards to dj/dt = -wz i, I understand that the change in j with respect to time is directly related to the yaw moment (wz) but what is the mathematical reasoning for using the i unit vector? Further, the 4th term demonstrates that dk/dt = -γ' / cos2 γ j, showing that k = -tanγ.
Sorry for the improper notation, was hoping to get further insight into how these terms are being derived.
 
Last edited:
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  • #2
I can't see your figure, and without that, there is no good way to reply to you. Please get the figure into the post.
 
  • #3
Dr.D said:
I can't see your figure, and without that, there is no good way to reply to you. Please get the figure into the post.
Sorry about that, should be fixed.
 

1. How is velocity calculated from the position vector in a rotating object?

Velocity can be calculated by taking the derivative of the position vector with respect to time. This is known as the instantaneous velocity and represents the speed and direction of the object at a specific moment in time.

2. Can the position vector and velocity change in a rotating object?

Yes, both the position vector and velocity can change in a rotating object. This is because the object is constantly moving and changing direction as it rotates.

3. What is the relationship between the position vector and velocity in a rotating object?

The position vector and velocity are always perpendicular to each other in a rotating object. This means that the position vector is tangent to the circular path of the object, while the velocity vector is always pointing towards the center of the circle.

4. How does the velocity from position vector differ from the linear velocity in a rotating object?

The velocity from position vector takes into account the changing direction of the object as it rotates, while linear velocity only considers the speed of the object in a straight line. Additionally, velocity from position vector is a vector quantity, meaning it has both magnitude and direction, while linear velocity is a scalar quantity with only magnitude.

5. How does the radius of rotation affect the velocity from position vector in a rotating object?

The radius of rotation has a direct effect on the velocity from position vector in a rotating object. The larger the radius, the greater the distance the object travels in one rotation, resulting in a higher velocity. Conversely, a smaller radius will result in a lower velocity.

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