Question

A 4m rope is tied between two supports with a tension of 500N. It is determined that the 5th harmonic mode of vibration occurs at 100Hz. What is the mass of the rope?

A) 0.25 kg
B) 0.5 kg
C) 1 kg
D) 2 kg

Answer 1

The mass of the rope is calculated using the formula for the nth harmonic frequency of a string, and the approximate mass of 0.3125 kg matches closest to option A) 0.25 kg.

Step-by-step explanation:

The question asks us to determine the mass of the rope based on its harmonic frequency and given tension. The formula relating the frequency (f) of the nth harmonic to the tension (T) and the linear mass density (μ) of a string is given by f = ⅛ n(⅛ T/μ) / 2L, where L is the length of the string and n is the harmonic number.

This can be rearranged to solve for μ as μ = n2T / 4L2f2. Since the mass of the rope (m) is the product of its linear mass density and its length (m = μ × L), we can substitute the given values (T = 500N, L = 4m, n = 5, f = 100Hz) to find m.

Substituting these values into the formula, we calculate:

μ = 52 × 500N / 4 × 4m2 × 100Hz2
μ = 2500 × 500 / 16 × 10000
μ = 0.078125 kg/m

Then, the mass of the rope is:
m = 0.078125 kg/m × 4m
m = 0.3125 kg

The closest answer to our calculation is 0.25 kg (Option A), since we typically round to the nearest given option in multiple-choice questions.