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 components of the motion
The equation can be separated into two components:
- \( x_1 = 6 \sin(10t) \)
- \( x_2 = 8 \cos(10t) \)

### Step 2: Recognize the amplitudes
From the components, we can identify the amplitudes:
- Amplitude of \( x_1 \) (denoted as \( A_1 \)) is 6 cm.
- Amplitude of \( x_2 \) (denoted as \( A_2 \)) is 8 cm.

### Step 3: Understand the phase difference
The sine and cosine functions have a phase difference of \( \frac{\pi}{2} \) radians (or 90 degrees). This means that the two components are perpendicular to each other.

### Step 4: Use the Pythagorean theorem to find the resultant amplitude
Since the two components are perpendicular, we can find the resultant amplitude \( A \) using the Pythagorean theorem:
\[ A = \sqrt{A_1^2 + A_2^2} \]

### Step 5: Substitute the values
Now, substituting the values of \( A_1 \) and \( A_2 \):
\[ A = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} \]

### Step 6: Calculate the resultant amplitude
Calculating the square root gives:
\[ A = 10 \text{ cm} \]

### Final Answer
The resultant amplitude of the simple harmonic motion is \( 10 \text{ cm} \).

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