To solve the problem step by step, we will follow the outlined approach in the video transcript.
**Step 1: Calculate the mass per unit length (µ) of the string.**
The mass of the string is given as \(1.0 \, \text{g}\), which can be converted to kilograms:
\[
1.0 \, \text{g} = 1.0 \times 10^{-3} \, \text{kg}
\]
The length of the string is \(20 \, \text{cm}\), which can be converted to meters:
\[
20 \, \text{cm} = 0.2 \, \text{m}
\]
Now, we can calculate the mass per unit length (µ):
\[
\mu = \frac{\text{mass}}{\text{length}} = \frac{1.0 \times 10^{-3} \, \text{kg}}{0.2 \, \text{m}} = 5.0 \times 10^{-3} \, \text{kg/m}
\]
**Step 2: Calculate the velocity (v) of the wave on the string.**
The velocity of the wave on the string can be calculated using the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where \(T\) is the tension in the string, which is given as \(0.5 \, \text{N}\).
Substituting the values:
\[
v = \sqrt{\frac{0.5 \, \text{N}}{5.0 \times 10^{-3} \, \text{kg/m}}}
\]
Calculating this gives:
\[
v = \sqrt{100} = 10 \, \text{m/s}
\]
**Step 3: Calculate the wavelength (λ) using the frequency (f).**
The frequency is given as \(100 \, \text{Hz}\). We can use the relationship between velocity, frequency, and wavelength:
\[
v = f \cdot \lambda \implies \lambda = \frac{v}{f}
\]
Substituting the values:
\[
\lambda = \frac{10 \, \text{m/s}}{100 \, \text{Hz}} = 0.1 \, \text{m}
\]
Converting this to centimeters:
\[
\lambda = 0.1 \, \text{m} = 10 \, \text{cm}
\]
**Step 4: Calculate the separation between successive nodes.**
The separation (D) between two successive nodes is given by:
\[
D = \frac{\lambda}{2}
\]
Substituting the value of λ:
\[
D = \frac{10 \, \text{cm}}{2} = 5 \, \text{cm}
\]
Thus, the separation between successive nodes on the string is \(5 \, \text{cm}\).
### Summary of the Solution:
The separation between two successive nodes on the string is **5 cm**.
