If equation of displacement of a particle `y=Asinpit+Bcospit` then motion of particle is

Options

A. Simple harmonic motion
B. Linear motion
C. Uniform circular motion
D. Uniform elliptical motion

Explanation

To determine the nature of the motion of a particle described by the displacement equation \( y = A \sin(\pi t) + B \cos(\pi t) \), we can follow these steps:

### Step 1: Identify the Displacement Equation
The displacement of the particle is given by:
\[ y = A \sin(\pi t) + B \cos(\pi t) \]

### Step 2: Find the Velocity
The velocity \( v \) is the rate of change of displacement with respect to time. We differentiate \( y \) with respect to \( t \):
\[ v = \frac{dy}{dt} = A \pi \cos(\pi t) - B \pi \sin(\pi t) \]

### Step 3: Find the Acceleration
The acceleration \( a \) is the rate of change of velocity with respect to time. We differentiate \( v \) with respect to \( t \):
\[ a = \frac{dv}{dt} = -A \pi^2 \sin(\pi t) - B \pi^2 \cos(\pi t) \]

### Step 4: Express Acceleration in Terms of Displacement
We can factor out \(-\pi^2\) from the acceleration equation:
\[ a = -\pi^2 (A \sin(\pi t) + B \cos(\pi t)) \]
Since \( A \sin(\pi t) + B \cos(\pi t) = y \), we can rewrite the acceleration as:
\[ a = -\pi^2 y \]

### Step 5: Determine the Nature of Motion
In simple harmonic motion (SHM), the acceleration is proportional to the displacement and is directed towards the mean position. The general form is:
\[ a = -k y \]
where \( k \) is a positive constant. Here, we have:
\[ a = -\pi^2 y \]
This confirms that the motion is indeed simple harmonic motion since the acceleration is proportional to the displacement.

### Conclusion
Thus, the motion of the particle described by the equation \( y = A \sin(\pi t) + B \cos(\pi t) \) is **simple harmonic motion (SHM)**.

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