Which of the following is not correct about a 20 g particle subjected to two SHM ,`x_1 = 2 sin 10t, `x_2 = 4 sin (10t + pi/3)` where `x_1 and x_2` are in metre and t is in second.

Options

A. The displacement of the particle at t=0 will be `2sqrt3` m.
B. Maximum speed of the particle will be `20sqrt7 m//s`.
C. Magnitude of maximum acceleration of the particle will be `200sqrt7 m//s^2`.
D. Energy of the resultant motion will be 26 J.

Explanation

To solve the problem, we need to analyze the two simple harmonic motions (SHMs) given by the equations:

1. \( x_1 = 2 \sin(10t) \)
2. \( x_2 = 4 \sin(10t + \frac{\pi}{3}) \)

where \( x_1 \) and \( x_2 \) are in meters and \( t \) is in seconds.

### Step 1: Identify the amplitudes of the SHMs
The amplitude of the first SHM, \( A_1 \), is 2 m, and the amplitude of the second SHM, \( A_2 \), is 4 m.

### Step 2: Calculate the resultant amplitude
The resultant amplitude \( A_r \) of two SHMs can be calculated using the formula:
\[ A_r = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]
where \( \phi \) is the phase difference between the two SHMs. Here, \( \phi = \frac{\pi}{3} \).

Calculating \( A_r \):
- \( A_1^2 = 2^2 = 4 \)
- \( A_2^2 = 4^2 = 16 \)
- \( \cos(\frac{\pi}{3}) = \frac{1}{2} \)

Now substituting these values into the formula:
\[ A_r = \sqrt{4 + 16 + 2 \cdot 2 \cdot 4 \cdot \frac{1}{2}} = \sqrt{4 + 16 + 8} = \sqrt{28} = 2\sqrt{7} \text{ m} \]

### Step 3: Calculate the maximum speed
The maximum speed \( v_{max} \) in SHM is given by:
\[ v_{max} = \omega A_r \]
where \( \omega \) is the angular frequency. From the equations, \( \omega = 10 \, \text{rad/s} \).

Now substituting the values:
\[ v_{max} = 10 \cdot 2\sqrt{7} = 20\sqrt{7} \, \text{m/s} \]

### Step 4: Calculate the maximum acceleration
The maximum acceleration \( a_{max} \) in SHM is given by:
\[ a_{max} = \omega^2 A_r \]

Calculating \( a_{max} \):
\[ a_{max} = 10^2 \cdot 2\sqrt{7} = 100 \cdot 2\sqrt{7} = 200\sqrt{7} \, \text{m/s}^2 \]

### Step 5: Calculate the total energy
The total energy \( E \) in SHM is given by:
\[ E = \frac{1}{2} k A_r^2 \]
where \( k \) is the spring constant. The spring constant \( k \) can be expressed in terms of mass \( m \) and angular frequency \( \omega \):
\[ k = m \omega^2 \]
Given that the mass \( m = 20 \, \text{g} = 0.02 \, \text{kg} \):
\[ k = 0.02 \cdot 10^2 = 0.02 \cdot 100 = 2 \, \text{N/m} \]

Now substituting into the energy formula:
\[ E = \frac{1}{2} \cdot 2 \cdot (2\sqrt{7})^2 = \frac{1}{2} \cdot 2 \cdot 28 = 28 \, \text{J} \]

### Conclusion
Now, we can evaluate the options based on the calculations:
1. Displacement - Correct
2. Maximum speed - Correct
3. Maximum acceleration - Correct
4. Total energy - Incorrect (calculated as 28 J, not 26 J)

Thus, the statement that is not correct is the one regarding the total energy.