Question

A body is executing simple harmonic motion. At a displacement \( x \), its potential energy is \( E_1 \), and at a displacement \( y \), its potential energy is \( E_2 \). The potential energy \( E \) at a displacement \( (x + y) \) is:

• 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. \(E = E_1 + E_2\)

Hint:

In simple harmonic motion, the potential energy follows a quadratic relation with displacement, leading to the sum of square roots property.

Answer 1

The Correct Option is B

Step 1: Understanding Potential Energy in SHM The potential energy in SHM is given by: \[ E = \frac{1}{2} k x^2 \] where \( k \) is the force constant. Step 2: Applying to Given Displacements For two different displacements: \[ E_1 = \frac{1}{2} k x^2, \quad E_2 = \frac{1}{2} k y^2 \] For total displacement \( (x+y) \): \[ E = \frac{1}{2} k (x+y)^2 \] Step 3: Using the Energy Sum Property Since energy follows a quadratic relationship: \[ \sqrt{E} = \sqrt{E_1} + \sqrt{E_2} \] Thus, the correct answer is option (2).

Answer 2

Given a body executing simple harmonic motion (SHM), the potential energy at displacement \( x \) is \( E_1 \), and at displacement \( y \) is \( E_2 \). Find the potential energy \( E \) at displacement \( (x + y) \).

Step 1: Recall that the potential energy in SHM is:
\[ E = \frac{1}{2} k A^2 \] where \( A \) is the displacement from mean position, and \( k \) is the spring constant.

Step 2: Potential energy is proportional to the square of displacement:
\[ E \propto x^2 \] So, \[ E_1 \propto x^2, \quad E_2 \propto y^2, \quad E \propto (x + y)^2 \]

Step 3: Write the relation:
\[ E = \frac{1}{2} k (x + y)^2 = \frac{1}{2} k (x^2 + y^2 + 2xy) \] \[ E = E_1 + E_2 + 2 \sqrt{E_1 E_2} \] because \( E_1 = \frac{1}{2} k x^2 \) and \( E_2 = \frac{1}{2} k y^2 \).

Step 4: Take square root of both sides:
\[ \sqrt{E} = \sqrt{E_1 + E_2 + 2 \sqrt{E_1 E_2}} = \sqrt{E_1} + \sqrt{E_2} \]

Therefore, the relation between \( E \), \( E_1 \), and \( E_2 \) is:
\[ \boxed{\sqrt{E} = \sqrt{E_1} + \sqrt{E_2}} \]