Deriving maximum velocity in a rotated frame

In summary, the conversation discusses how to derive the maximum velocity of a particle in the rotated uv plane. It is suggested to draw a diagram with a vector representing the maximum velocity and perpendicular lines to the u and v axes to find the coordinates in the rotated frame. The v coordinate of the vector may be negative in this example.
  • #1
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rotate%20frame_zps6lbjzvxf.jpg

I have a particle that travels in the cartesian plane with the maximum velocity of A units along the x-axis and B units along the y-axis per unit of time.

How do I go about deriving the maximum velocity of my particle in the rotated uv plane? (the maximum distance the particle can along the u and the v axes in one unit of time)

Any tips greatly appreciated.
 
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  • #2
Hi jumbo:

I suggest it might help if you add an arrow in the diagram to represent the vector M from the origin to point (x=A,y=B). Then draw perpendiculars from (A,B) to the axes u and v. If you know the angle between x-axis and u axis, you can calculate the length of the perpendiculars to the u axis and v axis which represent the M coordinates with respect to the v axis and u axis respectively.

Hope this helps.

Regards,
Buzz
 
  • #3
Thanks Buzz Bloom.

Is this what you mean?

vel_max_zpsuvcup2vg.jpg


Where the length of A' is the maximum distance my particle can travel along the u-axis of the rotated frame in one unit of time. Similarly, the length of B'...
 
Last edited:
  • #4
Hi jumbo:

I apologize for not expressing my thought more clearly.

You want to draw a line, call it L, from the origin to the point M. M will be on the vertical green line at the height B. L represents the vector corresponding to the maximum velocity of the particle you are describing.

Next draw lines from the point M perpendicular to the u and v axes. These lines will show the coordinates of the vector L in the (u,v) coordinate system. The resulting diagram should help you figure out how to calculate the u and v coordinates of the vector L.

By the way, for the particular example you have drawn, you may want to notice that the v coordinate of L is negative.

Hope this helps.

Regards,
Buzz
 
  • #5
I think I got it. Thanks Buzz

vel_max_2_zpscfsvfczk.jpg
 
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Likes Buzz Bloom

1. How is maximum velocity derived in a rotated frame?

The maximum velocity in a rotated frame is derived by using the principle of conservation of energy. This involves calculating the kinetic and potential energies in the rotated frame and then equating it to the total energy in the original frame.

2. What factors affect the maximum velocity in a rotated frame?

The maximum velocity in a rotated frame is affected by the magnitude and direction of the angular velocity of the frame, as well as the mass and shape of the object being rotated.

3. Can the maximum velocity in a rotated frame be greater than the maximum velocity in the original frame?

Yes, it is possible for the maximum velocity in a rotated frame to be greater than the maximum velocity in the original frame. This occurs when the angular velocity of the frame is in the same direction as the motion of the object.

4. How is the maximum velocity in a rotated frame related to the centripetal acceleration?

The maximum velocity in a rotated frame is directly related to the centripetal acceleration. In fact, the maximum velocity can be calculated by dividing the centripetal acceleration by the angular velocity of the frame.

5. Is it necessary to consider the rotation of the frame when calculating maximum velocity?

Yes, it is necessary to take into account the rotation of the frame when calculating the maximum velocity. The rotation of the frame affects the kinetic and potential energies of the object, which in turn determines the maximum velocity in the rotated frame.

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