A vibrating tuning fork is moving slowly and uniformly ins a horizontal circular path of radiu 8 m . The shortest distance of an observer ins ame plane from the tuning fork is 9m. The distance between the tuning fork and observer at the instant when apparent frequency becomes maximum is

Options

A. 9m
B. 25 m
C. 15 m
D. `sqrt(353)` m

Answer

To solve the problem, we need to find the distance between the tuning fork and the observer when the apparent frequency is maximum. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the setup
- The tuning fork is moving in a horizontal circular path with a radius of 8 m.
- The shortest distance from the observer to the tuning fork's circular path is 9 m.

### Step 2: Identify the positions
- Let’s denote:
- **O**: the position of the observer.
- **F**: the position of the tuning fork.
- **P**: the point on the circular path closest to the observer.

### Step 3: Calculate the distance OP
- The distance OP (from the observer to the closest point on the circular path) is given as 9 m.
- The radius of the circular path (the distance from the center of the circle to point P) is 8 m.

### Step 4: Determine the distance OF
- The distance OF (from the observer to the tuning fork) when the tuning fork is at point F (the position where the apparent frequency is maximum) is the sum of the shortest distance OP and the radius of the circular path:
\[ OF = OP + PF = 9 \, \text{m} + 8 \, \text{m} = 17 \, \text{m} \]

### Step 5: Use the Pythagorean theorem
- To find the distance between the observer and the tuning fork (distance OF) when the tuning fork is at point F, we can visualize a right triangle where:
- One leg is OP (9 m),
- The other leg is the radius (8 m).
- The hypotenuse (distance OF) can be calculated using the Pythagorean theorem:
\[ OF = \sqrt{OP^2 + PF^2} = \sqrt{9^2 + 8^2} \]
\[ OF = \sqrt{81 + 64} = \sqrt{145} \]

### Step 6: Final calculation
- Therefore, the distance between the tuning fork and the observer at the instant when the apparent frequency becomes maximum is:
\[ OF = \sqrt{145} \, \text{m} \]

### Conclusion
The distance between the tuning fork and the observer when the apparent frequency is maximum is \( \sqrt{145} \, \text{m} \).