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

Options

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

Answer

To find the value of epsilon for which the energy of both particles executing Simple Harmonic Motion (SHM) is the same, we can follow these steps:

### Step 1: Understand the energy of SHM
The total energy (E) of a particle in SHM is given by the formula:
\[ E = \frac{1}{2} k A^2 \]
where \( k \) is the force constant and \( A \) is the amplitude of the motion.

### Step 2: Relate energy to angular frequency
The force constant \( k \) can also be expressed in terms of mass \( m \) and angular frequency \( \omega \):
\[ k = m \omega^2 \]
Thus, the energy can be rewritten as:
\[ E = \frac{1}{2} m \omega^2 A^2 \]

### Step 3: Set up the equations for both particles
For the first particle:
- Displacement: \( x_1 = 4 \sin(10t + \frac{\pi}{6}) \)
- Amplitude \( A_1 = 4 \)
- Angular frequency \( \omega_1 = 10 \)

The energy of the first particle is:
\[ E_1 = \frac{1}{2} m (10^2) (4^2) \]

For the second particle:
- Displacement: \( x_2 = 5 \cos(\epsilon t) \)
- Amplitude \( A_2 = 5 \)
- Angular frequency \( \omega_2 = \epsilon \)

The energy of the second particle is:
\[ E_2 = \frac{1}{2} m (\epsilon^2) (5^2) \]

### Step 4: Set the energies equal
Since the energies of both particles are the same:
\[ E_1 = E_2 \]
This gives us:
\[ \frac{1}{2} m (10^2) (4^2) = \frac{1}{2} m (\epsilon^2) (5^2) \]

### Step 5: Simplify the equation
We can cancel \( \frac{1}{2} m \) from both sides (assuming \( m \neq 0 \)):
\[ (10^2)(4^2) = \epsilon^2 (5^2) \]

### Step 6: Substitute and solve for epsilon
Calculating the left side:
\[ 100 \times 16 = 1600 \]
Calculating the right side:
\[ \epsilon^2 \times 25 \]
Setting them equal gives:
\[ 1600 = 25 \epsilon^2 \]

Now, solving for \( \epsilon^2 \):
\[ \epsilon^2 = \frac{1600}{25} = 64 \]
Taking the square root:
\[ \epsilon = 8 \]

### Final Answer
The value of epsilon for which the energy of both particles is the same is:
\[ \epsilon = 8 \]

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