Chapter 3 Homework

Homework H3.F – Sp 26

8 Comments
Problem statement
Solution video

DISCUSSION THREAD


Any questions??

Discussion and hints:

It is RECOMMENDED that you choose to put your observer on the disk. In that case, the ω and α  that go into your acceleration equation will be that of the disk.

The discussion below follows a choice of putting the observer on the disk.

The disk shown above has TWO components of rotation:

  • a rotation rate of ω0 about a fixed axis (the “-” Z-axis), and,
  • a rotation rate of ωdisk about a moving axis (the “+” y-axis)

(Be sure to make a clear distinction between the lower case and upper case symbols.)

Therefore, the angular velocity of the disk is given by:

ω = -ω0K + ωdisk j

The angular acceleration vector α is simply the time derivative of the angular velocity vector ω : α = dω/dt. In taking this time derivative,

  • Recall that the K-axis is fixed. Since K is fixed, then dK/dt = 0.
  • Recall that the j-axis is NOT fixed. Knowing that, how do you find dj/dt?

 

8 thoughts on “Homework H3.F – Sp 26”

  1. Although it mentions relative velocity and the distances between points, I solved this problem with just MRF equations instead of full rigid-body formulas. Is this still a valid approach, or am I missing something?

    1. Not exactly sure of what you are asking. The MRF equations and the 3D rigid body equations are the same thing in this case. You are asked here to find the angular velocity (omega) and angular acceleration (alpha) which are used in both the MRF and rigid body equations. For this, you do not need relative velocity or relative acceleration terms, or distances between points.

      1. I believe Sydney was asking because the hint talked about using relative velocity and acceleration which is something I am also confused about. Are we meant to use those equations?

        1. The omega and alpha vectors requested in this problem are those that would be used if you were asked to find the acceleration of a point on this disk. In this particular version of the question, you were not asked to do that. The wording was chosen to help you put the problem in perspective with what we are doing in this topic of the course. I will remove those words in the Discussion above.

  2. If the observer is placed on the disk, the relative velocities of a to b would be zero correct? Because they are on that rotating frame. Would the velocity of point A in the external frame just be the addition of the angular velocities dotted with their respective distances from the points of rotation?

  3. Yes, on the first question. No, on the second question.

    For the first question you are correct in saying that since the observer is on the same reference frame as point A, A will appear to be stationary, both in velocity and acceleration.

    For the second question, you need to use the full MRF equation:
    v_A = v_B +(v_A/B)_rel + omega x r_A/B
    Here the velocity of B (v_B) is not zero: it moves in a circular path in the YZ-plane. Therefore, the velocity of A is a vector sum of that velocity with the omega x r_A/B vector.

    1

    1. To find the omega and alpha of the disk, you do not need to deal with point A. I think that the question asked here was just a question in general about using the velocity MRF equation.

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