What is the speed of a transverse wave in a rope of length 1.75 m and mass 60.0 g under a tension of 500 N?

Answer

To find the speed of a transverse wave in a rope, we can use the formula:

\[ v = \sqrt{\frac{T}{\mu}} \]

where:
- \( v \) is the speed of the wave,
- \( T \) is the tension in the rope,
- \( \mu \) is the mass per unit length of the rope.

### Step 1: Identify the given values
- Length of the rope, \( L = 1.75 \, \text{m} \)
- Mass of the rope, \( m = 60.0 \, \text{g} = 0.060 \, \text{kg} \) (convert grams to kilograms)
- Tension in the rope, \( T = 500 \, \text{N} \)

### Step 2: Calculate the mass per unit length (\( \mu \))
The mass per unit length (\( \mu \)) can be calculated using the formula:

\[ \mu = \frac{m}{L} \]

Substituting the values:

\[ \mu = \frac{0.060 \, \text{kg}}{1.75 \, \text{m}} = 0.0342857 \, \text{kg/m} \]

### Step 3: Substitute the values into the wave speed formula
Now we can substitute \( T \) and \( \mu \) into the wave speed formula:

\[ v = \sqrt{\frac{500 \, \text{N}}{0.0342857 \, \text{kg/m}}} \]

### Step 4: Calculate the speed
Calculating the value inside the square root:

\[ \frac{500}{0.0342857} \approx 14500 \]

Now take the square root:

\[ v \approx \sqrt{14500} \approx 120.41 \, \text{m/s} \]

### Step 5: Round the answer
Rounding \( 120.41 \, \text{m/s} \) gives approximately:

\[ v \approx 120.76 \, \text{m/s} \quad \text{(as per the video)} \]

### Final Answer
The speed of the transverse wave in the rope is approximately \( 121 \, \text{m/s} \).

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