A wire forming a loop is dipped into soap solution and taken out so that a film of soap solution is formed. A loop of 6.28 cm long thread is gently put on the film and the film is pricked with a needle inside the loop. The thread loop takes the shape of a circle. Find the tension in the thread. Surface tension of soap solution `=0.030Nm^-1`.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a loop of thread that is placed on a soap film. The length of the thread is given as 6.28 cm, and the surface tension of the soap solution is given as 0.030 N/m. The thread takes the shape of a circle when the film is pricked.

### Step 2: Convert the length of the thread to meters
The length of the thread is given in centimeters, so we need to convert it to meters for consistency in units.
\[ L = 6.28 \, \text{cm} = 6.28 \times 10^{-2} \, \text{m} \]

### Step 3: Relate the length of the thread to the radius of the circle
The length of the thread (L) is equal to the circumference of the circle formed by the thread. The formula for the circumference of a circle is:
\[ L = 2\pi r \]
where \( r \) is the radius of the circle. We can rearrange this to find the radius:
\[ r = \frac{L}{2\pi} \]

### Step 4: Calculate the radius
Substituting the value of \( L \):
\[ r = \frac{6.28 \times 10^{-2}}{2\pi} = \frac{6.28 \times 10^{-2}}{2 \times 3.14} = 1 \times 10^{-2} \, \text{m} = 0.01 \, \text{m} \]

### Step 5: Understand the forces acting on the thread
When the film is pricked, the surface tension acts along the edges of the film. The total upward force due to surface tension is balanced by the tension in the thread. The upward force due to surface tension can be expressed as:
\[ \text{Force due to surface tension} = \text{Surface Tension} \times \text{Length of the film} \]
Since the film has two surfaces (top and bottom), the effective length is \( 2r \):
\[ \text{Force due to surface tension} = S \times 2r \]

### Step 6: Set up the equation for tension
The tension \( T \) in the thread must balance the force due to surface tension:
\[ 2T = S \times 2r \]
This simplifies to:
\[ T = S \times r \]

### Step 7: Substitute the values to find tension
Now we can substitute the values of surface tension \( S = 0.030 \, \text{N/m} \) and radius \( r = 0.01 \, \text{m} \):
\[ T = 0.030 \times 0.01 = 0.0003 \, \text{N} = 3.0 \times 10^{-4} \, \text{N} \]

### Final Answer
The tension in the thread is:
\[ \boxed{3.0 \times 10^{-4} \, \text{N}} \]