A particle is executing SHM with an amplitude 4 cm. the displacment at which its energy is half kinetic and half potential is

Options

A. 1 cm
B. `2^(1//2) cm`
C. `2 cm`
D. `2(2)^(1//2) cm`

Explanation

To solve the problem of finding the displacement at which the kinetic energy and potential energy of a particle executing simple harmonic motion (SHM) are equal, we can follow these steps:

### Step 1: Understand the Energy in SHM
In simple harmonic motion, the total mechanical energy (E) of the system is the sum of kinetic energy (KE) and potential energy (PE). When the energies are equal, we have:
\[ KE = PE = \frac{E}{2} \]

### Step 2: Write the Expressions for Kinetic and Potential Energy
The expressions for kinetic energy and potential energy in SHM are:
- Kinetic Energy:
\[ KE = \frac{1}{2} k (A^2 - x^2) \]
- Potential Energy:
\[ PE = \frac{1}{2} k x^2 \]
Where:
- \( k \) is the spring constant,
- \( A \) is the amplitude,
- \( x \) is the displacement from the mean position.

### Step 3: Set the Energies Equal
Since we want to find the displacement \( x \) where \( KE = PE \), we can set the two expressions equal to each other:
\[ \frac{1}{2} k (A^2 - x^2) = \frac{1}{2} k x^2 \]

### Step 4: Simplify the Equation
We can cancel \( \frac{1}{2} k \) from both sides (assuming \( k \neq 0 \)):
\[ A^2 - x^2 = x^2 \]

### Step 5: Rearrange the Equation
Rearranging gives us:
\[ A^2 = 2x^2 \]

### Step 6: Substitute the Amplitude
Given that the amplitude \( A = 4 \) cm, we substitute this value into the equation:
\[ 4^2 = 2x^2 \]
\[ 16 = 2x^2 \]

### Step 7: Solve for \( x^2 \)
Dividing both sides by 2:
\[ x^2 = \frac{16}{2} = 8 \]

### Step 8: Find \( x \)
Taking the square root of both sides gives us:
\[ x = \sqrt{8} = 2\sqrt{2} \text{ cm} \]

### Conclusion
The displacement at which the kinetic energy and potential energy are equal is:
\[ x = 2\sqrt{2} \text{ cm} \]

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