Calculate the ratio of displacement to amplitude when kinetic energy of a body is thrice its potential energy.

Explanation

To solve the problem, we need to calculate the ratio of displacement (x) to amplitude (A) when the kinetic energy (KE) of a body is thrice its potential energy (PE).

### Step-by-Step Solution:

1. **Understanding the relationship between KE and PE**:
We are given that the kinetic energy is three times the potential energy:
\[ KE = 3 \times PE \]

2. **Using the law of conservation of energy**:
The total mechanical energy (E) in simple harmonic motion (SHM) is the sum of kinetic energy and potential energy:
\[ E = KE + PE \]
Substituting the expression for KE from step 1:
\[ E = 3 \times PE + PE = 4 \times PE \]

3. **Expressing potential energy in terms of total energy**:
From the equation above, we can express potential energy in terms of total energy:
\[ PE = \frac{E}{4} \]

4. **Using the formula for potential energy in SHM**:
The potential energy (PE) in SHM is given by:
\[ PE = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

5. **Expressing total energy in terms of amplitude**:
The total energy (E) in SHM is given by:
\[ E = \frac{1}{2} k A^2 \]
where \( A \) is the amplitude.

6. **Setting the expressions for potential energy equal**:
Now we can set the expression for potential energy from step 4 equal to the expression from step 3:
\[ \frac{1}{2} k x^2 = \frac{E}{4} \]
Substituting the expression for total energy from step 5 into this equation:
\[ \frac{1}{2} k x^2 = \frac{1}{4} \left(\frac{1}{2} k A^2\right) \]

7. **Simplifying the equation**:
This simplifies to:
\[ \frac{1}{2} k x^2 = \frac{1}{8} k A^2 \]
Dividing both sides by \(\frac{1}{2} k\):
\[ x^2 = \frac{1}{4} A^2 \]

8. **Taking the square root**:
Taking the square root of both sides gives:
\[ x = \frac{A}{2} \]

9. **Calculating the ratio of displacement to amplitude**:
The ratio of displacement to amplitude is:
\[ \frac{x}{A} = \frac{\frac{A}{2}}{A} = \frac{1}{2} \]

### Final Answer:
The ratio of displacement to amplitude when the kinetic energy is thrice the potential energy is:
\[ \frac{x}{A} = \frac{1}{2} \]