A body is executing simple harmonic motion As `x` displacement `x` its potential energy is `E_(1)` and at a displacement `y` its potential energy is `E_(2)` The potential energy `E` at displacement `(x+y)` is

Options

A. `sqrt(E ) = sqrt(E_(1)) - sqrt(E_(2))`
B. `sqrt(E ) = sqrt(E_(1)) + sqrt(E_(2))`
C. `E = E_(1)+E_(2)`
D. `E = E_(1)+E_(2)`

Explanation

To solve the problem of finding the potential energy \( E \) at a displacement \( (x+y) \) given the potential energies \( E_1 \) and \( E_2 \) at displacements \( x \) and \( y \) respectively, we can follow these steps:

### Step 1: Understand the formula for potential energy in SHM
The potential energy \( E \) in simple harmonic motion (SHM) is given by the formula:
\[ E = \frac{1}{2} k x^2 \]
where \( k \) is a constant related to the system (specifically \( k = m \omega^2 \), with \( m \) being mass and \( \omega \) being angular frequency).

### Step 2: Write expressions for \( E_1 \) and \( E_2 \)
Using the formula for potential energy, we can express \( E_1 \) and \( E_2 \) as:
\[ E_1 = \frac{1}{2} k x^2 \]
\[ E_2 = \frac{1}{2} k y^2 \]

### Step 3: Solve for \( x \) and \( y \) in terms of \( E_1 \) and \( E_2 \)
From the equations for \( E_1 \) and \( E_2 \), we can express \( x \) and \( y \) as follows:
\[ x = \sqrt{\frac{2E_1}{k}} \]
\[ y = \sqrt{\frac{2E_2}{k}} \]

### Step 4: Find the total displacement \( (x+y) \)
Now, we need to find the potential energy at the displacement \( (x+y) \):
\[ E = \frac{1}{2} k (x+y)^2 \]

### Step 5: Substitute \( x \) and \( y \) into the equation
Substituting the expressions for \( x \) and \( y \):
\[ E = \frac{1}{2} k \left( \sqrt{\frac{2E_1}{k}} + \sqrt{\frac{2E_2}{k}} \right)^2 \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ E = \frac{1}{2} k \left( \frac{2E_1}{k} + \frac{2E_2}{k} + 2 \sqrt{\frac{2E_1}{k} \cdot \frac{2E_2}{k}} \right) \]
\[ E = \frac{1}{2} k \cdot \frac{2}{k} \left( E_1 + E_2 + 2 \sqrt{E_1 E_2} \right) \]
\[ E = E_1 + E_2 + 2 \sqrt{E_1 E_2} \]

### Step 7: Final result
Thus, the potential energy \( E \) at displacement \( (x+y) \) is:
\[ E = E_1 + E_2 + 2 \sqrt{E_1 E_2} \]