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Thread: static friction of masses connected by a rod

  1. #1 static friction of masses connected by a rod 
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    This isn't homework...I'm reviewing physics after many years of neglect.

    Given 2 masses, , connected by a rigid, massless rod, stationary with respect to a ramp which makes an angle of with the horizontal, with coefficients of static friction between the masses and the ramp = respectively, what is the magnitude of the tension or compression in the rod, and what are the magnitudes of the static friction, , acting on each mass?

    This assumes is small enough that the masses do not lose traction, i.e.,



    Note that if the masses are assumed to already be moving, then the problem is straightforward:





    Where T is the tension in the rod (T<0 implies compression). Such problems are common in basic physics, e.g., Halliday, Resnick, & Krane, 4th Ed., chap.6, problem 29.

    So I assumed it would be straightforward to do the same problem, but with the masses stationary. But I get 2 equations (sum of the forces for each mass) and 3 unknowns (T, ).

    I find it surprising that such a simple situation is undetermined, and assume I've missed something simple.

    Note: I've looked through several physics texts and can find only problems in which the masses are already moving.

    Also, it's driving me crazy!


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  3. #2  
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    Quote Originally Posted by inkliing View Post
    This isn't homework...I'm reviewing physics after many years of neglect.

    Given 2 masses, , connected by a rigid, massless rod, stationary with respect to a ramp which makes an angle of with the horizontal, with coefficients of static friction between the masses and the ramp = respectively, what is the magnitude of the tension or compression in the rod, and what are the magnitudes of the static friction, , acting on each mass?

    This assumes is small enough that the masses do not lose traction, i.e.,



    Note that if the masses are assumed to already be moving, then the problem is straightforward:





    Where T is the tension in the rod (T<0 implies compression). Such problems are common in basic physics, e.g., Halliday, Resnick, & Krane, 4th Ed., chap.6, problem 29.

    So I assumed it would be straightforward to do the same problem, but with the masses stationary. But I get 2 equations (sum of the forces for each mass) and 3 unknowns (T, ).

    I find it surprising that such a simple situation is undetermined, and assume I've missed something simple.

    Note: I've looked through several physics texts and can find only problems in which the masses are already moving.

    Also, it's driving me crazy!
    Don't feel bad, this situation is quite common in Statics and in Material Resistance. The solution is to assume an infinitesimal movement of the system and the problem becomes determinate.


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  4. #3  
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    The problem could have been stated more clearly. At first I was imagining the masses side by side on the ramp instead of one above the other. It's not surprising that the solution is indeterminate if you consider that the masses could be connected by a slack string, and wouldn't go anywhere, or they could be connected by a stretched rubber band and wouldn't go anywhere either, until the rubber band is stretched enough to make one of the masses start sliding.
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  5. #4  
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    Thx for the helpful responses.


    It now seems clear to me that, since rigid rods do not exist, the rod's strain determines its tension or compression, leaving 2 equations (sum of forces on the masses) and the 2 unknown static frictions.
    In other words, if the angle in the ramp is small enough such that both masses, when standing alone with no rod connecting them, experience static frictional forces strictly less than their respective maximums, then, in general, at the same angle, when the masses are then connected by a flexible rod, the rod will experience a nonzero tension or compression resulting in a nonzero tensile or compressive strain of the rod. If this strain is known then the system is determined.
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