Homework H2.I - Sp23

Problem statement
Solution video

Any questions??


DISCUSSION
In some sense, this is a very standard kinematics of rigid bodies problem. A rigid link connects points A and B. To relate the motion of these two points, you will need the following kinematics equations:

vA = vB + ω x rA/B 
aA = aBα x rA/B - ω2 rA/B

The nuance in this problem is the acceleration of point A. From what we learned earlier in the semester, the acceleration of a point can be written in terms of its path coordinates; that is, here the acceleration of A can be resolved into its tangential and normal components. As you proceed on this problem, you first need to recognize the tangential and normal unit vectors, et and en, for the motion of A. Then write down the acceleration of A, first in its path components, and then in its Cartesian component. In the end, you will have two scalar equations coming from the rigid body acceleration equation in terms of two unknowns.

26 thoughts on “Homework H2.I - Sp23”

    1. You need to relate the motion of A and B through the rigid body kinematics equations. Use the fact that you know the path of A. That is, you know the direction of the velocity for A, and you know the directions for the tangential and normal components of acceleration of A. Balance out the i and j components of the acceleration equation relating A and B, and using the path description for A.

    2. Try using the other definition of acceleration provided to us at the beginning of the semester that states that acceleration is the sum of the rate of change of speed (v_dot) and velocity divided by rho.

    1. Sam: As state in the above DISCUSSION, your kinematics equation will produce two scalar equations with two unknonwns. Those two unknowns are the angular acceleration of AB and v_dot for A. Solve those two equations together for your answers.

    1. No, I believe that you should use the kinematic equations given underneath the problem in the Discussion portion.
      vA = vB + ω x rA/B
      aA = aB + α x rA/B - ω2 rA/B

    1. I'm also getting really large numbers. I'm hoping that this is just because the units are mm and the given parameters are in the hundreds because other than that I'm pretty confident in my processes so I think you're probably okay on that.

  1. Once we solve for the Cartesian acceleration equations, are we able to plug them straight into vectors et and en? I am thinking this because et is purely horizontal and en is purely vertical, so theoretically wouldn't those two vectors equal the i and j vectors from Cartesian?

    1. It feels like I should be able to, but I am not sure what I would set the equations equal to. I think both the i and j components of acceleration are non-zero, and I am not sure what other values can be put in to solve the two equations.

      1. When working with the acceleration equation for link AB:

        aA = aB + α x rA/B - ω2 rA/B

        you need to specify what you know about aA and aB.

        For aB, it is a known value aB in a direction that is aligned with the ramp.

        For aA, you know that it has two components: one in the e_t direction (the unknown, vA_dot) and one in the e__n direction (the known, vA^2/rho).

        Substitute both of those expressions for aA and aB into the acceleration equation above, and balance out your i and j components. This gives you two scalar equations in terms of two unknowns: vA_dot and alpha_AB. Solve for those two unknowns.

  2. I've written out both my acceleration equations, but I'm not sure how to relate the two of them. Do I use the angle we are given to convert the path equations to cartesian with trig functions?

    1. Peter: Not sure of the acceleration equations to which you refer here. You have really only one acceleration equation relating the motions of A and B. In that equation, you need to provide expressions for aA and aB (the latter is zero, as you have pointed out earlier). This gives you one vector equation, which becomes two scalar equations. Solve for the unknowns. Please read the detailed response that was given earlier, and repeated below.
      ______________________________

      When working with the acceleration equation for link AB:

      aA = aB + α x rA/B - ω2 rA/B

      you need to specify what you know about aA and aB.

      For aB, it is a known value aB in a direction that is aligned with the ramp.

      For aA, you know that it has two components: one in the e_t direction (the unknown, vA_dot) and one in the e__n direction (the known, vA^2/rho).

      Substitute both of those expressions for aA and aB into the acceleration equation above, and balance out your i and j components. This gives you two scalar equations in terms of two unknowns: vA_dot and alpha_AB. Solve for those two unknowns.

  3. At the top of the arc path that point A follows, the acceleration components of A would be exactly vertical and horizontal in Path and Cartesian coordinates. So, is finding expressions for i and j and equating them to the expressions for et and en the correct method? How should I handle v_dot and v^2 in the path expressions? If they're the same as v_A and a_A, it seems like I would end up going in equation circles.

    1. I am not sure if this is right but what I did was get my e_t and e_n equations set up and then I said that V^2/rho = wAB^2*R because w=v/r --> w^2=v^2/r^2. This allowed me to solve for alpha AB but i got a very small number for my angular acceleration.

      1. I did something similar to this. I set e^n equal to the j vector of acceleration so I had aAj = Va^2/P=pe^n. I believe that for this problem the j and e^n vectors are equivalent considering the position of particle A which allows us to equate these two vectors.

      2. The way I saw it, e_t is in the same direction as v_A (i_hat) because they are both tangent to the semicircular path. This made e_n point in the -j_hat direction. Using this, I broke my kinematic accel. equation ( aA = aB + α x rA/B - ω2 rA/B ) into i and j, and then replaced i and j with vA_dot and vA^2/rho, respectively. This allowed me to solve for alpha and vA_dot.

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