A body executing `S.H.M`. has its velocity `10cm//s` and `7 cm//s` when its displacement from the mean positions are `3 cm` and `4 cm` respectively. Calculate the length of the path.

Explanation

To solve the problem step by step, we will use the relationship between velocity, displacement, and amplitude in simple harmonic motion (SHM).

### Step 1: Understand the relationship in SHM
In SHM, the velocity \( v \) at a displacement \( x \) from the mean position is given by the formula:
\[ v = \omega \sqrt{A^2 - x^2} \]
where:
- \( v \) is the velocity,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude,
- \( x \) is the displacement from the mean position.

### Step 2: Set up the equations for the two given conditions
We have two conditions:
1. When \( x_1 = 3 \, \text{cm} \), \( v_1 = 10 \, \text{cm/s} \)
2. When \( x_2 = 4 \, \text{cm} \), \( v_2 = 7 \, \text{cm/s} \)

Using the formula for each condition, we can write:
\[ v_1 = \omega \sqrt{A^2 - x_1^2} \quad \text{(1)} \]
\[ v_2 = \omega \sqrt{A^2 - x_2^2} \quad \text{(2)} \]

### Step 3: Divide the two equations
Dividing equation (1) by equation (2):
\[ \frac{v_1}{v_2} = \frac{\sqrt{A^2 - x_1^2}}{\sqrt{A^2 - x_2^2}} \]
Substituting the values:
\[ \frac{10}{7} = \frac{\sqrt{A^2 - 3^2}}{\sqrt{A^2 - 4^2}} \]
This simplifies to:
\[ \frac{10}{7} = \frac{\sqrt{A^2 - 9}}{\sqrt{A^2 - 16}} \]

### Step 4: Square both sides
Squaring both sides gives:
\[ \left(\frac{10}{7}\right)^2 = \frac{A^2 - 9}{A^2 - 16} \]
Calculating the left side:
\[ \frac{100}{49} = \frac{A^2 - 9}{A^2 - 16} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying yields:
\[ 100(A^2 - 16) = 49(A^2 - 9) \]
Expanding both sides:
\[ 100A^2 - 1600 = 49A^2 - 441 \]
Rearranging gives:
\[ 100A^2 - 49A^2 = 1600 - 441 \]
\[ 51A^2 = 1159 \]
Thus:
\[ A^2 = \frac{1159}{51} \]
Calculating \( A^2 \):
\[ A^2 \approx 22.75 \]
Taking the square root:
\[ A \approx \sqrt{22.75} \approx 4.76 \, \text{cm} \]

### Step 6: Calculate the length of the path
The length of the path in SHM is given by:
\[ \text{Length of path} = 2A \]
Substituting the value of \( A \):
\[ \text{Length of path} = 2 \times 4.76 \approx 9.52 \, \text{cm} \]

### Final Answer
The length of the path is approximately \( 9.52 \, \text{cm} \).

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