Question

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.)

Answer 1

2.56 m/s²

Step-by-step 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