In a simple harmonic motion

Options

A. the maximum potential energy equals the maximum kinetic energy
B. the minimum potential energy equals the minimum kinetic energy
C. the minimum potential energy equals the maximum kinetic energy
D. the maximum potential energy equals the minimum kinetic energy.

Answer

To solve the question regarding simple harmonic motion (SHM) and determine the correct options from the given statements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Energy in SHM**:
- In simple harmonic motion, a particle oscillates back and forth around a mean position. The energy of the particle can be divided into two types: kinetic energy (KE) and potential energy (PE).
- The formulas for kinetic energy (KE) and potential energy (PE) in SHM are:
- \( KE = \frac{1}{2} m \omega^2 (A^2 - x^2) \)
- \( PE = \frac{1}{2} m \omega^2 x^2 \)
- Here, \( m \) is the mass of the particle, \( \omega \) is the angular frequency, \( A \) is the amplitude, and \( x \) is the displacement from the mean position.

2. **Finding Maximum Kinetic Energy**:
- The maximum kinetic energy occurs when the displacement \( x = 0 \) (at the mean position).
- Substituting \( x = 0 \) into the kinetic energy formula:
\[ KE_{\text{max}} = \frac{1}{2} m \omega^2 A^2 \]

3. **Finding Maximum Potential Energy**:
- The maximum potential energy occurs when the displacement \( x = A \) (at the extreme positions).
- Substituting \( x = A \) into the potential energy formula:
\[ PE_{\text{max}} = \frac{1}{2} m \omega^2 A^2 \]

4. **Comparing Maximum Energies**:
- From the above calculations, we see that:
\[ KE_{\text{max}} = PE_{\text{max}} = \frac{1}{2} m \omega^2 A^2 \]
- Therefore, **Option 1** is correct: Maximum potential energy equals maximum kinetic energy.

5. **Finding Minimum Kinetic Energy**:
- The minimum kinetic energy occurs at the extreme position where \( x = A \).
- Substituting \( x = A \) into the kinetic energy formula:
\[ KE_{\text{min}} = \frac{1}{2} m \omega^2 (A^2 - A^2) = 0 \]

6. **Finding Minimum Potential Energy**:
- The minimum potential energy occurs at the mean position where \( x = 0 \).
- Substituting \( x = 0 \) into the potential energy formula:
\[ PE_{\text{min}} = \frac{1}{2} m \omega^2 (0^2) = 0 \]

7. **Comparing Minimum Energies**:
- From the above calculations, we find that:
\[ KE_{\text{min}} = 0 \quad \text{and} \quad PE_{\text{min}} = 0 \]
- Therefore, **Option 2** is also correct: Minimum potential energy equals minimum kinetic energy.

8. **Conclusion**:
- Since we have established that Options 1 and 2 are correct, we can conclude that Options 3 and 4 are incorrect.

### Final Answer:
- Correct Options: **1 and 2**