Two SHW are represented by the equations `x_1 = 20 sin [5pit +pi/4] and x_2 = 10 (sin5pit+sqrt(3) cos 5 pit]` . The ratio of the amplitudes of the two motions is

Options

A. 0.5
B. 1
C. 0.25
D. `sqrt3/2`

Answer

To solve the problem, we need to find the amplitudes of the two simple harmonic motions (SHM) given by the equations:

1. \( x_1 = 20 \sin(5 \pi t + \frac{\pi}{4}) \)
2. \( x_2 = 10 \left( \sin(5 \pi t) + \sqrt{3} \cos(5 \pi t) \right) \)

### Step 1: Identify the amplitude of \( x_1 \)

The equation for \( x_1 \) is already in the standard form of SHM, which is \( A \sin(\omega t + \phi) \). Here, the amplitude \( A_1 \) is simply the coefficient of the sine function.

\[ A_1 = 20 \]

### Step 2: Rewrite \( x_2 \) in standard form

The equation for \( x_2 \) is not in standard form because it contains both sine and cosine terms. To find the amplitude, we need to combine these terms into a single sine function.

We can use the formula:

\[ R \sin(\omega t + \phi) = A \sin(\omega t) + B \cos(\omega t) \]

where \( R = \sqrt{A^2 + B^2} \) and \( \tan(\phi) = \frac{B}{A} \).

In our case:
- \( A = 10 \)
- \( B = 10\sqrt{3} \)

### Step 3: Calculate \( R \)

Now we calculate \( R \):

\[ R = \sqrt{(10)^2 + (10\sqrt{3})^2} \]
\[ = \sqrt{100 + 300} \]
\[ = \sqrt{400} \]
\[ = 20 \]

So, the amplitude \( A_2 \) of \( x_2 \) is:

\[ A_2 = 20 \]

### Step 4: Find the ratio of the amplitudes

Now that we have both amplitudes, we can find the ratio of the amplitudes \( \frac{A_1}{A_2} \):

\[ \frac{A_1}{A_2} = \frac{20}{20} = 1 \]

Thus, the ratio of the amplitudes of the two motions is:

\[ 1 : 1 \]

### Final Answer

The ratio of the amplitudes of the two motions is \( 1 : 1 \).

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