The ratio of maximum acceleration to maximum velocity in a simple harmonic motion is `10 s^(-1)` At, t = 0 the displacement is 5 m. What is the maximum acceleration ? The initial phase is `(pi)/(4)`

Options

A. 500` m//s^(2)`
B. `750 sqrt(2) m//s^(2)`
C. `750 m//s^(2)`
D. `500sqrt(2) m//s^(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have the following data:
- The ratio of maximum acceleration to maximum velocity in simple harmonic motion (SHM) is \(10 \, \text{s}^{-1}\).
- The displacement at \(t = 0\) is \(5 \, \text{m}\).
- The initial phase is \(\frac{\pi}{4}\).

### Step 2: Write the equations for SHM
In SHM, the displacement \(x\) can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
where:
- \(A\) is the amplitude,
- \(\omega\) is the angular frequency,
- \(\phi\) is the initial phase.

### Step 3: Substitute the initial conditions
At \(t = 0\), the displacement is given as \(5 \, \text{m}\):
\[ x(0) = A \sin\left(\frac{\pi}{4}\right) = 5 \]
Since \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), we can write:
\[ A \cdot \frac{1}{\sqrt{2}} = 5 \]
Thus, solving for \(A\):
\[ A = 5\sqrt{2} \, \text{m} \]

### Step 4: Relate maximum acceleration and maximum velocity
The maximum acceleration \(A_{\text{max}}\) and maximum velocity \(V_{\text{max}}\) in SHM are given by:
\[ A_{\text{max}} = \omega^2 A \]
\[ V_{\text{max}} = \omega A \]

### Step 5: Set up the ratio
We know from the problem that:
\[ \frac{A_{\text{max}}}{V_{\text{max}}} = 10 \]
Substituting the expressions for maximum acceleration and maximum velocity:
\[ \frac{\omega^2 A}{\omega A} = 10 \]
This simplifies to:
\[ \omega = 10 \, \text{s}^{-1} \]

### Step 6: Calculate maximum acceleration
Now, we can substitute \(\omega\) and \(A\) into the equation for maximum acceleration:
\[ A_{\text{max}} = \omega^2 A = (10)^2 \cdot (5\sqrt{2}) = 100 \cdot 5\sqrt{2} \]
Calculating this gives:
\[ A_{\text{max}} = 500\sqrt{2} \, \text{m/s}^2 \]

### Final Answer
The maximum acceleration is:
\[ A_{\text{max}} = 500\sqrt{2} \, \text{m/s}^2 \]