At `x = (A)/(4)`, what fraction of the mechanical energy is potential ? What fraction is kinetic ? Assume potential energy to be zero at mean position.

Explanation

To solve the problem, we need to find the fractions of potential and kinetic energy when the displacement \( x \) is equal to \( \frac{A}{4} \), where \( A \) is the amplitude of the motion. We will assume that the potential energy is zero at the mean position.

### Step-by-Step Solution:

1. **Understanding Total Mechanical Energy**:
The total mechanical energy \( E \) in simple harmonic motion (SHM) is given by:
\[ E = \frac{1}{2} k A^2 \]
where \( k \) is the spring constant and \( A \) is the amplitude.

2. **Calculating Potential Energy at \( x = \frac{A}{4} \)**:
The potential energy \( U \) at a displacement \( x \) in SHM is given by:
\[ U = \frac{1}{2} k x^2 \]
Substituting \( x = \frac{A}{4} \):
\[ U = \frac{1}{2} k \left(\frac{A}{4}\right)^2 = \frac{1}{2} k \cdot \frac{A^2}{16} = \frac{k A^2}{32} \]

3. **Expressing Potential Energy in Terms of Total Energy**:
We can express the potential energy as a fraction of the total energy:
\[ U = \frac{k A^2}{32} \]
Since the total energy \( E = \frac{1}{2} k A^2 \), we can write:
\[ \text{Fraction of potential energy} = \frac{U}{E} = \frac{\frac{k A^2}{32}}{\frac{1}{2} k A^2} = \frac{1/32}{1/2} = \frac{1}{16} \]

4. **Calculating Kinetic Energy**:
The kinetic energy \( K \) can be found using the relationship:
\[ K = E - U \]
Substituting the values we have:
\[ K = \frac{1}{2} k A^2 - \frac{k A^2}{32} \]
To combine these, we need a common denominator:
\[ K = \frac{16 k A^2}{32} - \frac{k A^2}{32} = \frac{15 k A^2}{32} \]

5. **Expressing Kinetic Energy in Terms of Total Energy**:
Now, we can express the kinetic energy as a fraction of the total energy:
\[ \text{Fraction of kinetic energy} = \frac{K}{E} = \frac{\frac{15 k A^2}{32}}{\frac{1}{2} k A^2} = \frac{15/32}{1/2} = \frac{15}{16} \]

### Final Results:
- The fraction of mechanical energy that is potential at \( x = \frac{A}{4} \) is \( \frac{1}{16} \) or \( 6.25\% \).
- The fraction of mechanical energy that is kinetic at \( x = \frac{A}{4} \) is \( \frac{15}{16} \) or \( 93.75\% \).