The S.H.M. of a particle is given by the equation `y=3 sin omegat + 4 cosomega t` . The amplitude is

Options

A. 7
B. 1
C. 5
D. 12

Answer

To find the amplitude of the simple harmonic motion represented by the equation \( y = 3 \sin(\omega t) + 4 \cos(\omega t) \), we can follow these steps:

### Step 1: Identify the components of the equation
The given equation is:
\[ y = 3 \sin(\omega t) + 4 \cos(\omega t) \]
Here, the coefficients of the sine and cosine functions are 3 and 4, respectively.

### Step 2: Combine the sine and cosine terms
To express the equation in the standard form of simple harmonic motion, we can combine the sine and cosine terms into a single sine function with a phase shift. This can be done using the formula:
\[ R \sin(\omega t + \phi) = R \sin(\omega t) \cos(\phi) + R \cos(\omega t) \sin(\phi) \]
where \( R \) is the amplitude and \( \phi \) is the phase angle.

### Step 3: Calculate the amplitude \( R \)
From the equation, we can identify:
\[ R \cos(\phi) = 3 \quad \text{and} \quad R \sin(\phi) = 4 \]
To find \( R \), we can use the Pythagorean theorem:
\[ R = \sqrt{(R \cos(\phi))^2 + (R \sin(\phi))^2} \]
Substituting the values, we have:
\[ R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Conclusion
Thus, the amplitude of the simple harmonic motion is:
\[ \text{Amplitude} = 5 \]

### Summary
The amplitude of the particle executing simple harmonic motion described by the equation \( y = 3 \sin(\omega t) + 4 \cos(\omega t) \) is 5.

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