To find the wavelength of the mode of vibration with the lowest frequency (the fundamental mode) for a string fixed at both ends, we can use the following principles:

1. Fundamental Mode Wavelength:

   • For a string fixed at both ends, the wavelength $\lambda$ of the fundamental frequency is twice the length of the string. $\lambda = 2L$ where $L$ is the length of the string.

2. Given Data:

   • Length of the string, $L = 35.0 \, \text{cm} = 0.35 \, \text{m}$.

3. Calculate the Wavelength:

   • Plugging in the length: $\lambda = 2 \times 0.35 \, \text{m} = 0.7 \, \text{m}$

So, the wavelength of the mode of vibration with the lowest frequency is 0.7 m.

Answer: The correct choice is 0.7 m.

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Would you like a further explanation on the principles behind this? Here are five related questions that expand on this topic:

1. How would the wavelength change if the string length were increased to 50 cm?
2. What role does the tension in the string play in determining the frequency of vibration?
3. How would the frequency of the fundamental mode change if the string’s mass per unit length were doubled?
4. Why is the wavelength of the fundamental mode always twice the length of the string in this scenario?
5. How can you calculate the speed of a wave on a string given tension and mass per unit length?

Tip: Remember, for strings fixed at both ends, the fundamental wavelength is always twice the string's length.