In an experiment of standing waves, a string `90 cm` long is attached to the prong of an electrically driven tuning fork that scillates perpendicular to the length of the string at a frequency of `60 Hz`. The mass of the string is `0.044 kg`. What tension (in newton) must the string be under (weights are attached to the other end) if it is to oscillate in four loops?

Answer

To solve the problem step by step, we will follow the principles of standing waves and wave motion on a string.

### Step 1: Determine the wavelength of the standing wave
In a standing wave with `n` loops (or antinodes), the length of the string can be expressed as:
\[ L = n \frac{\lambda}{2} \]
Given that the string is `90 cm` long (which is `0.9 m`) and it oscillates in `4 loops` (n = 4), we can set up the equation:
\[ 0.9 = 4 \frac{\lambda}{2} \]
This simplifies to:
\[ 0.9 = 2\lambda \]
Thus, we can find the wavelength:
\[ \lambda = \frac{0.9}{2} = 0.45 \, \text{m} \]

### Step 2: Calculate the wave velocity
The wave velocity \( v \) on the string can be calculated using the formula:
\[ v = f \cdot \lambda \]
Where:
- \( f = 60 \, \text{Hz} \) (frequency)
- \( \lambda = 0.45 \, \text{m} \)

Substituting the values:
\[ v = 60 \cdot 0.45 = 27 \, \text{m/s} \]

### Step 3: Determine the mass per unit length (linear density) of the string
The mass of the string is given as `0.044 kg` and the length of the string is `0.9 m`. The linear density \( \mu \) is calculated as:
\[ \mu = \frac{m}{L} = \frac{0.044 \, \text{kg}}{0.9 \, \text{m}} \approx 0.04889 \, \text{kg/m} \]

### Step 4: Calculate the tension in the string
The tension \( T \) in the string can be calculated using the formula:
\[ T = \mu v^2 \]
Substituting the values we have:
\[ T = 0.04889 \cdot (27)^2 \]
Calculating \( (27)^2 \):
\[ (27)^2 = 729 \]
Now substituting this back into the tension formula:
\[ T = 0.04889 \cdot 729 \approx 35.6 \, \text{N} \]

### Conclusion
Thus, the tension that must be applied to the string for it to oscillate in four loops is approximately:
\[ \boxed{36 \, \text{N}} \]