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Newton's Second Law (Translational Dynamics)

AP Physics 1· difficulty 4/5

L θ

A small bob of mass mm swings in a horizontal circle on a string of length LL that makes angle θ\theta with the vertical. What is the period TT of revolution?

  • A

    T=2πLtan⁡θ/gT = 2\pi\sqrt{L\tan\theta/g}

  • B

    T=2πLsin⁡θ/gT = 2\pi\sqrt{L\sin\theta/g}

  • C

    T=2πLcos⁡θ/gT = 2\pi\sqrt{L\cos\theta/g}

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  • D

    T=2πL/gT = 2\pi\sqrt{L/g}

Explanation

Vertical: Tscos⁡θ=mgT_s\cos\theta = mg. Horizontal: Tssin⁡θ=mω2rT_s\sin\theta = m\omega^2 r with r=Lsin⁡θr = L\sin\theta. Dividing yields ω2=g/(Lcos⁡θ)\omega^2 = g/(L\cos\theta), so period T=2πLcos⁡θ/gT = 2\pi\sqrt{L\cos\theta/g}.

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