Two strings (1) and (2) of equal thickness are made up of the same material. The length of the first string (1) is half that of the second string (2), while tension is (1) is twice that in the string, (2). Compare the velocities of the transverse waves on them.

Answer

To compare the velocities of the transverse waves on the two strings, we can follow these steps:

### Step 1: Identify the parameters for both strings
Let:
- \( L_1 \) = length of string 1
- \( L_2 \) = length of string 2
- \( T_1 \) = tension in string 1
- \( T_2 \) = tension in string 2
- \( \mu_1 \) = mass per unit length of string 1
- \( \mu_2 \) = mass per unit length of string 2

Given:
- \( L_1 = \frac{1}{2} L_2 \)
- \( T_1 = 2 T_2 \)
- \( \mu_1 = \mu_2 \) (since both strings are of equal thickness and made of the same material)

### Step 2: Write the formula for the velocity of transverse waves
The velocity \( v \) of transverse waves on a string is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension and \( \mu \) is the mass per unit length.

### Step 3: Calculate the velocities for both strings
For string 1:
\[ v_1 = \sqrt{\frac{T_1}{\mu_1}} \]
For string 2:
\[ v_2 = \sqrt{\frac{T_2}{\mu_2}} \]

### Step 4: Set up the ratio of the velocities
To compare the velocities, we can set up the ratio:
\[ \frac{v_1}{v_2} = \frac{\sqrt{T_1 / \mu_1}}{\sqrt{T_2 / \mu_2}} = \sqrt{\frac{T_1}{T_2}} \cdot \sqrt{\frac{\mu_2}{\mu_1}} \]

### Step 5: Substitute the known values
Since \( \mu_1 = \mu_2 \), the ratio \( \frac{\mu_2}{\mu_1} = 1 \). Therefore, the equation simplifies to:
\[ \frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}} \]

### Step 6: Substitute the tension values
We know that \( T_1 = 2 T_2 \). Substituting this into the equation gives:
\[ \frac{v_1}{v_2} = \sqrt{\frac{2 T_2}{T_2}} = \sqrt{2} \]

### Step 7: Conclusion
Thus, we can conclude that:
\[ v_1 = \sqrt{2} v_2 \]
This means that the velocity of transverse waves on string 1 is \( \sqrt{2} \) times the velocity on string 2.

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