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, we need to find the length of the path of a body executing Simple Harmonic Motion (S.H.M) given its velocities at specific displacements. We will follow these steps:

### Step 1: Write the equation for velocity in S.H.M.
The velocity \( V \) of a body in S.H.M can be expressed as:
\[ V = \sqrt{A^2 - x^2} \cdot \omega \]
where \( A \) is the amplitude, \( x \) is the displacement from the mean position, and \( \omega \) is the angular frequency.

### Step 2: Set up the equations for the two given conditions.
From the problem, 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 velocity equation, we can write:
\[ V_1 = \sqrt{A^2 - x_1^2} \cdot \omega \quad \text{(1)} \]
\[ V_2 = \sqrt{A^2 - x_2^2} \cdot \omega \quad \text{(2)} \]

### Step 3: Divide the two equations to eliminate \( \omega \).
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:
\[ \frac{10}{7} = \frac{\sqrt{A^2 - 3^2}}{\sqrt{A^2 - 4^2}} \]

### Step 4: Square both sides to eliminate the square roots.
\[ \left(\frac{10}{7}\right)^2 = \frac{A^2 - 9}{A^2 - 16} \]
This simplifies to:
\[ \frac{100}{49} = \frac{A^2 - 9}{A^2 - 16} \]

### Step 5: Cross-multiply to solve for \( A^2 \).
Cross-multiplying gives:
\[ 100(A^2 - 16) = 49(A^2 - 9) \]
Expanding both sides:
\[ 100A^2 - 1600 = 49A^2 - 441 \]
Rearranging gives:
\[ 100A^2 - 49A^2 = 1600 - 441 \]
\[ 51A^2 = 1159 \]
\[ A^2 = \frac{1159}{51} \]
Calculating \( A^2 \):
\[ A^2 \approx 22.75 \quad \Rightarrow \quad A \approx \sqrt{22.75} \approx 4.76 \, \text{cm} \]

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

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

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