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Lagrangian mechanics, simple pendulum

  1. Apr 15, 2016 #1
    1. The problem statement, all variables and given/known data
    A simple pendulum of length ξ and mass m is suspended from a point on the circumference of a thin massless disc of radius α that rotates with a constant angular velocity ω about its central axis as shown in Figure. Find the equation of motion of the mass m.
    Untitled.jpg
    2. Relevant equations
    L = T - V

    3. The attempt at a solution
    Firstly, I know I should find its x' and y' which representing its velocity.
    i.e. T - V = 1/2 m ( x_dot 2 + y_dot 2 ) - mgy

    However, I don't know where I should start with to find their x and y.

    Thanks.
     
  2. jcsd
  3. Apr 15, 2016 #2
    first task would be to define the generalized coordinates q and velocities qdots and then proceed for defining the T and V.
    your disk is in which plane and which is axis of rotation of the disk.
     
  4. Apr 15, 2016 #3
    But how can I define the generalized coordinates q and qdots?
     
  5. Apr 15, 2016 #4
    define usual coordinates as degrees of freedom permits -write any constraining equations...which are relations between coordinates or velocities.
    then the gen. coordinates can be defined.
    see your text book
     
  6. Apr 15, 2016 #5
    your pendulum is hanging from a disk- so it will be at length L from the disk suppose you place your origin of coordinates at the centre of disk
    and the axes X,Y,Z so the bob will lie at -z,x,y but as disl starts rotating the bob will start rotating in a circle and that circle will be a raised one , so at any instant the bob will be at x',y'z' .
    the equation of constraint can be that sum of the squares of three coordinates will be equal to length square + rad of the disk squared.
    if you choose an angle made by the thread with vertical the cosine of the angle will be z' /L.... similarly other relations can follow and your degrees of freedom will be reduced- the motion may be described by one angle and its time rate of change.
     
  7. Apr 16, 2016 #6
    I try to begin the question with place origin of coordinates at the centre of the disk (x,y)
    With the parameters of figures given,
    for x, because ξsinθ ( the height of triangle formed by bob ) is longer than that of radius,
    x = ξsinθ - a cos ωt
    similarly, y = ξcosθ - a sin ωt

    However, the answer is x = a cos ωt + ξ sin θ, y = a sin ωt - ξcosθ
    Am I missing something?

    If I just begin the problem that the disk rotates on the other side, I get the same answer as textbook.
    Last night, I spent a night to watch youtube mechanics and vector teaching videos and read over my textbook but still I could not get the answer.
    I am new to Mechanics. sorry for any annoying questions :'(
     
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