# Free vibrations of stretched strings

• kaamos
In summary, the problem involves a stretched string driven by two sources with the same frequency and amplitude, but 180 degrees out of phase. The question is asking for the smallest normal mode frequency of the string. The solution involves the formula ω=π(T/LM)^(1/2) and the attempt involves differentiating both y1(x,t) and y2(x,t) with respect to t and x.
kaamos
A stretched string of mass m, length L, and tension T is driven by two
sources, one at each end. The sources both have the same frequency  and
amplitude A, but are exactly 180 degrees out of phase with respect to one another.
(Each end is an antinode). What is the smallest normal mode frequency of the
string?

Solution: ω=π(T/LM)^(1/2)

Attempt: y1(x,t)=f(x)cosωt and y2(x,t)=f(x)cosωt
Then differentiating both of them w.r.t t and x. Am I even on the right track??

By the way, I'm guessing this problem is from an intro course, so I've moved it to the introductory physics forum.

## 1. What are free vibrations of stretched strings?

Free vibrations of stretched strings refer to the natural oscillations that occur when a string is plucked or struck and then allowed to vibrate freely without any external force or energy input.

## 2. What factors affect the free vibrations of stretched strings?

The factors that affect the free vibrations of stretched strings include the tension of the string, its length, mass per unit length, and the material properties of the string, such as its density and stiffness.

## 3. How do you calculate the frequency of free vibrations in a stretched string?

The frequency of free vibrations in a stretched string can be calculated using the equation: f = (1/2L) * √(T/μ), where L is the length of the string, T is the tension, and μ is the mass per unit length.

## 4. What is the relationship between the frequency and wavelength of free vibrations in a stretched string?

The frequency and wavelength of free vibrations in a stretched string are inversely proportional. This means that as the frequency increases, the wavelength decreases, and vice versa.

## 5. How do boundary conditions affect the free vibrations of stretched strings?

The boundary conditions, such as fixed or free ends, can affect the mode of vibration and the frequency of free vibrations in a stretched string. For example, a string with both ends fixed will have different modes of vibration and frequencies than a string with one end fixed and the other end free.

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