The displacement of two identical particles executing `SHM` are represented by equations `x_(1) = 4 sin (10t+(pi)/(6)) & x_(2) = 5 cos (omegat)` For what value of `•`, energy of both the particles is same.

Options

A. 16 unit
B. 6 unit
C. 4 unit
D. 8 unit

Explanation

To solve the problem of finding the value of \( \omega \) for which the energies of two identical particles executing simple harmonic motion (SHM) are the same, we will follow these steps:

### Step 1: Identify the equations of motion
The equations of motion for the two particles are given as:
- \( x_1 = 4 \sin(10t + \frac{\pi}{6}) \)
- \( x_2 = 5 \cos(\omega t) \)

### Step 2: Extract the amplitudes and angular frequencies
From the equations, we can identify:
- For \( x_1 \):
- Amplitude \( A_1 = 4 \)
- Angular frequency \( \omega_1 = 10 \)

- For \( x_2 \):
- Amplitude \( A_2 = 5 \)
- Angular frequency \( \omega_2 = \omega \)

### Step 3: Write the expression for energy in SHM
The energy \( E \) of a particle in SHM is given by the formula:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
where \( m \) is the mass of the particle, \( \omega \) is the angular frequency, and \( A \) is the amplitude.

### Step 4: Calculate the energy for both particles
- For particle 1 (from \( x_1 \)):
\[ E_1 = \frac{1}{2} m \omega_1^2 A_1^2 = \frac{1}{2} m (10^2) (4^2) = \frac{1}{2} m \cdot 100 \cdot 16 = 800m \]

- For particle 2 (from \( x_2 \)):
\[ E_2 = \frac{1}{2} m \omega_2^2 A_2^2 = \frac{1}{2} m \omega^2 (5^2) = \frac{1}{2} m \omega^2 \cdot 25 = \frac{25}{2} m \omega^2 \]

### Step 5: Set the energies equal to each other
Since we want the energies to be equal, we set \( E_1 = E_2 \):
\[ 800m = \frac{25}{2} m \omega^2 \]

### Step 6: Cancel the mass \( m \) from both sides
Assuming \( m \neq 0 \), we can cancel \( m \):
\[ 800 = \frac{25}{2} \omega^2 \]

### Step 7: Solve for \( \omega^2 \)
Multiply both sides by 2 to eliminate the fraction:
\[ 1600 = 25 \omega^2 \]
Now, divide both sides by 25:
\[ \omega^2 = \frac{1600}{25} = 64 \]

### Step 8: Take the square root to find \( \omega \)
\[ \omega = \sqrt{64} = 8 \]

### Final Answer
Thus, the value of \( \omega \) for which the energies of both particles are the same is:
\[ \omega = 8 \text{ units} \]

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