A mass m oscillates with simple harmonic motion with frequency `f = (omega)/(2 pi)` and amplitude A on a spring of stiffness constant K. Which of the following is not correct ?

Options

A. The frequency is `(1)/(2pi) sqrt((K)/(m))`
B. The maximum velocity occurs when x = 0
C. The total energy of the system is `(1)/(2) KA^(2)`
D. Its minimum potential energy occurs when x = A

Answer

To solve the question, we need to analyze the properties of a mass oscillating in simple harmonic motion (SHM) on a spring. We are given the frequency, amplitude, and spring constant, and we need to determine which statement about the system is not correct.

### Step-by-Step Solution:

1. **Understanding the Basics of SHM**:
- A mass \( m \) attached to a spring oscillates with simple harmonic motion (SHM). The frequency \( f \) is given by the formula:
\[ f = \frac{\omega}{2\pi} \]
- The angular frequency \( \omega \) is related to the spring constant \( K \) and mass \( m \) by:
\[ \omega = \sqrt{\frac{K}{m}} \]

2. **Equations of Motion**:
- The displacement \( x \) of the mass from the mean position can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
- The maximum velocity \( v_{\text{max}} \) occurs when \( x = 0 \) and is given by:
\[ v_{\text{max}} = A \omega \]

3. **Energy in SHM**:
- The total mechanical energy \( E \) in SHM is the sum of kinetic energy (KE) and potential energy (PE):
\[ E = \text{KE} + \text{PE} \]
- The potential energy stored in the spring when it is displaced by \( x \) is:
\[ \text{PE} = \frac{1}{2} K x^2 \]
- The kinetic energy when the mass is moving with velocity \( v \) is:
\[ \text{KE} = \frac{1}{2} m v^2 \]

4. **Total Energy Calculation**:
- The total energy can also be expressed in terms of amplitude \( A \):
\[ E = \frac{1}{2} K A^2 \]
- This energy remains constant throughout the motion.

5. **Analyzing Potential Energy**:
- The potential energy is maximum when the displacement \( x \) is at its maximum (i.e., \( x = A \) or \( x = -A \)).
- Conversely, the potential energy is minimum (zero) when \( x = 0 \) (the mean position).

6. **Identifying the Incorrect Statement**:
- The question asks which statement is not correct. If a statement claims that minimum potential energy occurs at \( x = A \), this is incorrect because:
- At \( x = A \) (maximum displacement), the potential energy is maximum, not minimum.

### Conclusion:
The statement that "minimum potential energy occurs at \( x = A \)" is incorrect. Therefore, the answer to the question is that the option stating this is the one that is not correct.