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27 April, 00:51

If the phase angle for a block-spring system in SHM is ϕ and the block's position is given by x = xm cos (ωt + ϕ), what is the ratio of the kinetic energy to the potential energy at time t = 0? (Use any variable or symbol stated above as necessary.)

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  1. 27 April, 02:24
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    K. E/P. E = m/k tan²φ x ω²

    Explanation:

    The given position of block x = x₀ cos (ωt + φ)

    The velocity of block v = dx/dt = - x₀ sin (ωt + φ) x ω

    The kinetic energy = 1/2 mv² = 1/2 m x₀² sin² (ωt + φ) x ω²

    The potential energy of spring = 1/2 k x², where k is the spring constant

    Thus P. E = 1/2 x k x x₀² cos² (ωt + φ)

    When t = 0

    K. E = 1/2 m x₀²sin²φ x ω²

    P. E = 1/2 k x₀² cos²φ

    Dividing these, we have

    K. E/P. E = m/k tan²φ x ω²
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