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Math Help - Collisions and Many Particle Systems

  1. #1
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    Collisions and Many Particle Systems

    Is anybody able to help out with the attached questions please?
    Attached Files Attached Files
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  2. #2
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    How much have you managed to complete?

    Tell me where you are stuck and I can help you. I'd feel uncomfortable just giving a set of answers to your homework.
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  3. #3
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    I can't do any of part (a). My mind just goes blank when I look at this question.

    Moreover, this is not homework. I'm trying to study mechanics by myself and this is a question I found without solutions.
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  4. #4
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    OK, for a (i) can you see how the lengths are related, right?

    From the diagram you should be able to see that

    \boxed{y+l = d + x}

    by equating the distances that get you from the wall to the mass m_1 .

    ii. &iii. What are the salient features here:

    1. The string over the pulley transmits the same tension force, T to each mass.
    2. The fact that m_1 and m_2 are connected in this way means they must have the same acceleration.
    3. The table is smooth so there's no friction forces on m_2 .

    So for the mass m_1 you must have:

    \boxed{m_1 a = m_1 g - T} [1]

    where g is the acceleration due to gravity and a is the inertial acceleration. This is from Newton II.

    Using Newton II for mass m_2 we must have:

    \boxed{m_2 a = T - F_s} [2]

    where F_s is the force due to the extension of the spring.

    I hope the diagrams should be obvious from the above. As for the pulley the forces you have are the tension T acting away from the pulley at each corner and reaction force diagonally outwards from the pulley (at a 45 degree angle to the horizontal).

    The only force equation left is that of the spring where:

    \boxed{F_s = k (y- l_0 )} [3].

    Sub [3] into [2] and you get:

    \boxed{m_2 a = T - k( y - l_0 )} [4]

    and now add [1] to [4] to give you

    ( m_1 + m_2 ) a = m_1 g - k ( y - l_0 ),

    rearranging this and substituting for y gives us:

    \boxed{a = \frac{m_1 g - k (d+x -l - l_0 )}{m_1 + m_2}} .

    Hope that was of some help!
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