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The balance wheel of a watch oscillates with angular amplitude 1.7π rad and period 0.64 s. Find (a) the maximum angular speed of the wheel, (b) the angular speed of the wheel at displacement 1.7π/2 rad, and (c) the magnitude of the angular acceleration at displacement 1.7π/4 rad.

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kylie

  1. It undergoes simple harmonic motion with an equation for angular displacement, a(t), that can be written

    a(t) = A sin (2*pi*t/P)
    where A is the amplitude and p is the period. I chose time zero such that the phase angle is zero at that time, to simplify the equation.

    (a) w = A*(2*pi/P) cos(2*pi*t/P)
    w(max) = 2*pi*(1.7 pi)/(0.46)

    (b) When the displacement is half the amplitude,
    2*pi*t/P = pi/6 radians
    At that time, w is cos (pi/6) = (sqrt3)/2 times the amplitude

    (c) When the displacement is 1/4 the amplitude, the angular argument
    2*pi*t/P = 0.253 radians
    The angular acceleration at any time is
    alpha (t) = -A*(2*pi/P)^2*sin(2*pi*t/P)
    = (A/4)(2*pi/P)^2

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    drwls

  2. for part a: is it the 2*PI*(1.7PI)/.46 OR IS IT 2*pi* (1.7pi/.46)? AND FOR PART B:
    DO YOU MULTIPLY THE AMPLITUDE BY (SQRT3)/2? AND I UNDERSTAND PART C

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    kylie

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