The equation of a simple harmonic motion is given by `x =6 sin 10 t + 8 cos 10 t`, where x is in cm, and t is in seconds. Find the resultant amplitude.

Answer

To find the resultant amplitude of the simple harmonic motion given by the equation \( x = 6 \sin(10t) + 8 \cos(10t) \), we can follow these steps:

### Step 1: Identify the amplitudes
The equation consists of two components:
- The first component \( x_1 = 6 \sin(10t) \) has an amplitude \( a_1 = 6 \) cm.
- The second component \( x_2 = 8 \cos(10t) \) has an amplitude \( a_2 = 8 \) cm.

### Step 2: Use the formula for resultant amplitude
The resultant amplitude \( A \) for two simple harmonic motions can be calculated using the formula:
\[ A = \sqrt{a_1^2 + a_2^2 + 2 a_1 a_2 \cos(\Delta \phi)} \]
where \( \Delta \phi \) is the phase difference between the two components.

### Step 3: Determine the phase difference
The sine and cosine functions have a phase difference of \( \frac{\pi}{2} \) radians (or 90 degrees). Therefore, we have:
\[ \Delta \phi = \frac{\pi}{2} \]

### Step 4: Calculate \( \cos(\Delta \phi) \)
Using the value of the phase difference:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]

### Step 5: Substitute values into the formula
Now substitute \( a_1 = 6 \), \( a_2 = 8 \), and \( \cos(\Delta \phi) = 0 \) into the formula:
\[ A = \sqrt{6^2 + 8^2 + 2 \cdot 6 \cdot 8 \cdot 0} \]
This simplifies to:
\[ A = \sqrt{36 + 64 + 0} \]
\[ A = \sqrt{100} \]

### Step 6: Final calculation
Calculating the square root gives:
\[ A = 10 \text{ cm} \]

### Conclusion
The resultant amplitude of the simple harmonic motion is \( 10 \) cm.

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