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9 July, 13:16

A thin hoop is hung on a wall, supported by a horizontal nail. The hoop's mass is M=2.0 kg and its radius is R=0.6 m. What is the period of small oscillations of the hoop? Type your answer below, accurate to two decimal places, and assuming it is in seconds. [Recall that the moment of inertia of a hoop around its center is Icm=MR2.]

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  1. 9 July, 13:36
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    Given that,

    Mass of the thin hoop

    M = 2kg

    Radius of the hoop

    R = 0.6m

    Moment of inertial of a hoop is

    I = MR²

    I = 2 * 0.6²

    I = 0.72 kgm²

    Period of a physical pendulum of small amplitude is given by

    T = 2π √ (I / Mgd)

    Where,

    T is the period in seconds

    I is the moment of inertia in kgm²

    I = 0.72 kgm²

    M is the mass of the hoop

    M = 2kg

    g is the acceleration due to gravity

    g = 9.8m/s²

    d is the distance from rotational axis to center of of gravity

    Therefore, d = r = 0.6m

    Then, applying the formula

    T = 2π √ (I / MgR)

    T = 2π √ (0.72 / (2 * 9.8 * 0.6)

    T = 2π √ (0.72 / 11.76)

    T = 2π √0.06122

    T = 2π * 0.2474

    T = 1.5547 seconds

    T ≈ 1.55 seconds to 2d•p

    Then, the period of oscillation is 1.55seconds
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