1-if you decrease the hanging mass by a factor of four( 0.55 kg/4) but keep the linear mass density the same, by what factor does the fundamental frequency change. Explain using equations. .2-if you decrease the linear mass density by a factor of four, but keep the hanging mass same, by what factor does the fundamental frequency change. Explain using equations.The correct answer and explanation is:
Explanation and Answer:
The fundamental frequency of a string is governed by the equation: f1=12LTμf_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}
where:
- f1f_1 is the fundamental frequency,
- LL is the length of the string (constant in this case),
- TT is the tension in the string,
- μ\mu is the linear mass density (mass per unit length\text{mass per unit length}).
The tension TT in the string is caused by the hanging mass and is given by: T=mgT = m g
where:
- mm is the hanging mass,
- gg is the acceleration due to gravity.
Part 1: Decreasing the Hanging Mass by a Factor of Four
If mm is decreased by a factor of 4 (m′=m4m’ = \frac{m}{4}), the tension TT is also reduced by a factor of 4 since T=mgT = m g. Substituting this into the frequency equation: f1′=12LT′μ=12LT4μ=12LT4μf_1′ = \frac{1}{2L} \sqrt{\frac{T’}{\mu}} = \frac{1}{2L} \sqrt{\frac{\frac{T}{4}}{\mu}} = \frac{1}{2L} \sqrt{\frac{T}{4\mu}}
Taking out the factor of 14=12\frac{1}{\sqrt{4}} = \frac{1}{2}: f1′=f12f_1′ = \frac{f_1}{2}
Answer for Part 1: The fundamental frequency decreases by a factor of 2.
Part 2: Decreasing the Linear Mass Density by a Factor of Four
If μ\mu is decreased by a factor of 4 (μ′=μ4\mu’ = \frac{\mu}{4}), the tension TT remains constant because the hanging mass mm is unchanged. Substituting this into the frequency equation: f1′=12LTμ′=12LTμ4=12L4Tμf_1′ = \frac{1}{2L} \sqrt{\frac{T}{\mu’}} = \frac{1}{2L} \sqrt{\frac{T}{\frac{\mu}{4}}} = \frac{1}{2L} \sqrt{\frac{4T}{\mu}}
Taking out the factor of 4=2\sqrt{4} = 2: f1′=2f1f_1′ = 2 f_1
Answer for Part 2: The fundamental frequency increases by a factor of 2.
Summary
- Decreasing the hanging mass by a factor of 4 decreases the fundamental frequency by a factor of 2.
- Decreasing the linear mass density by a factor of 4 increases the fundamental frequency by a factor of 2.
This happens because the fundamental frequency depends on the square root of the tension and inversely on the square root of the linear mass density. Adjusting these parameters directly influences the vibration characteristics of the string.