26 thoughts on “Homework H5.A - Sp24”

  1. What equation is used to relate the crate's acceleration with the pulley's angular acceleration? I don't think we can use the rigid body kinematics because we don't know anything about the pulley's angular speed

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    1. The acceleration of the crate at the point where is cable is attached is the same as the tangential acceleration of the pulley around which the cable is wrapped. The angular speed of the pulley will influence the normal component of acceleration of that point on the pulley, but not the tangential acceleration. Therefore, you do not need to know the angular velocity of the pulley.

      Does this help?

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  2. Can we assume the normal force on point A be 0 when it is on the verge of tipping? Can we factor that into the Euler equation?

    1. That would not be an assumption, but rather a fact. Yes, you need to set the normal force at A to zero to find the maximum alpha.

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      1. Would this be just for the euler equation? Or can we also set normal force at A equal to zero in the summation of forces equation?

  3. When doing the moment equation should we include friction for both points or only for point b?

    1. In general, there is a friction force and a normal force at both feet A and B. At some point in your analysis, you will set the normal force at A equal to zero in order to satisfy the requirement for impending tipping. With that normal force going to zero, the friction force at A will also be zero.

    1. The radius is used to relate the translational acceleration to the angular acceleration.

  5. So when we write the sum of the forces and the euler equation and cancel out Na because it will be zero when it tips. For the tension force we don't know it yet but we can rewrite it using the sum of forces equations right? That way the moment equation is in terms of variables we know and acceleration in the x. Is this the right way to think about it?

  6. The best way to think of this is that the Newton/Euler equations provide us with THREE equations: summation of forces in x, summation of forces in y, and summation of moments about G.

    These three equations have three unknowns: a_Gx, F_T (the tension force on the crate by the cable), and the normal force at B, N_B. (Note that the friction force at B is f_B = muK*N_B, the angular acceleration of the crate is zero, and, as you state, N_A = f_A = 0.)

    Solve these three equations for the three unknowns. From a_Gx, you can find the angular acceleration of the pulley needed to have the crate of the verge of tipping.

    1. To do this, would we use the rigid body equation for acceleration? If so, do we consider the angular velocity to be zero?

  7. In my simplifying, I am left with g, and not mg, so can I assume g to just be 9.81? Or is the answer supposed to be different so a value of 120lbs is used in the calculation?

  8. I was able to create a system of equations solving for both ma and T, which I was then able to relate the solved variables into the angular acceleration and tension.

  9. When solving the Euler equation, notice that points A and B are neither centers of mass nor fixed. So rather than finding the sum of the moments at each point, the Euler equation must be taken about the center point of the crate G.

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  10. I was able to use equations for the sum of forces in the x and y directions to solve for the linear acceleration. From there, I could use this value in the sum of moments equation to solve for T. Don't forget to use ~32 for g since units are in feet.

  11. Because the object is already moving/slipping, you can't use the tipping method from statics. You need to use a summation of forces instead. Also a = r*alpha will be necessary.

    1. The method is almost the exact same, aside from the fact that the horizontal acceleration is no longer zero.

      The normal force at A will be equal to zero since we are looking at maximum force, or the force just before tipping. We then come up with our 3 kinetics equations, the sum of forces in the two directions and the sum of the moments about the center of gravity. Because we are solving for acceleration in x in order to find angular acceleration, there is one additional step to the usual method of slipping/tipping that we saw in Statics.

  12. When I took the moment of the crate about point B, the force required in the cable to tip the crate is significantly less than when I take the moment of the crate about its center of mass. Is the reason for this because of Euler's equation? Or is there something else causing this?

    1. If you use the short form of Euler's equation ( sum M = I*alpha), you cannot use point B as a reference point because B is an accelerating point. Taking moments about point B will give you an incorrect answer.

      If G is your reference point, then you can use the short form of the equation.

  13. For this problem, it is very helpful to realize that the total angular momentum of the block is 0. This will thus help you set your moment equation to be equal to 0. For this equation, I highly recommend taking it around the center of mass, so your equation simplifies the easiest.

  14. When setting up the kinematics of this problem remember that at the instant before the crate tips, the normal force at A is zero. This also will mean that the friction force at that point will be zero.

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