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17 June, 12:04

An astronaut on a small planet wishes to measure the local value of g by timing pulses traveling down a wire which has a large object suspended from it. Assume a wire of mass 4.30 g is 1.60 m long and has a 3.00-kg object suspended from it. A pulse requires 59.9 ms to traverse the length of the wire. Calculate gplanet from these data. (You may neglect the mass of the wire when calculating the tension in it.)

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  1. 17 June, 14:17
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    2.56 m/s²

    Explanation:

    A standing wave is produced in the wire, its frequency f = n/2l√ (T/μ). For the fundamental frequency, n = 1.

    f = 1/2l√ (T/μ)

    where l = length of wire = 1.60 m, T₁ = tension in wire = weight of object = m₁g (neglecting wires mass), m₁ = mass of object = 3.00 kg, g = acceleration due to gravity on the small planet, μ = linear density of wire = m₀/l, m₀ = mass of wire = 4.30 g = 0.0043 kg and f = 1/T where T = period of pulse = 59.9 ms = 0.0599 s

    f = 1/2l√ (T₀/μ) = 1/T ⇒ T₁ = 4l²μ/T²

    m₁g = 4l²μ/T²

    g = 4l²μ/m₁T² = 4l²m₀/l/m₁T² = 4lm₀/m₁T²

    g = 4lm₀/m₁T² = 4 * 1.60 * 0.0043 / (3.00 * 0.0599²) = 2.56 m/s²
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