Two simple harmonic motions given by, `x = a sin (omega t+delta)` and `y = a sin (omega t + delta + (pi)/(2))` act on a particle will be

Options

A. circular anti-clockwise
B. circular clockwise
C. elliptical anti-clockwise
D. elliptical clockwise

Answer

To solve the problem of the two simple harmonic motions acting on a particle, we will analyze the equations given and derive the resultant motion step by step.

### Step 1: Identify the equations of motion
The two simple harmonic motions are given by:
1. \( x = a \sin(\omega t + \delta) \)
2. \( y = a \sin(\omega t + \delta + \frac{\pi}{2}) \)

### Step 2: Simplify the second equation
Using the trigonometric identity, we know that:
\[ \sin(\theta + \frac{\pi}{2}) = \cos(\theta) \]
Thus, we can rewrite the second equation:
\[ y = a \cos(\omega t + \delta) \]

### Step 3: Express both equations in terms of sine and cosine
Now we have:
1. \( x = a \sin(\omega t + \delta) \)
2. \( y = a \cos(\omega t + \delta) \)

### Step 4: Find the relationship between \( x \) and \( y \)
We can use the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Substituting the expressions for \( x \) and \( y \):
\[ \left(\frac{x}{a}\right)^2 + \left(\frac{y}{a}\right)^2 = \sin^2(\omega t + \delta) + \cos^2(\omega t + \delta) = 1 \]
This simplifies to:
\[ x^2 + y^2 = a^2 \]

### Step 5: Interpret the result
The equation \( x^2 + y^2 = a^2 \) represents a circle of radius \( a \) in the \( xy \)-plane. This indicates that the motion of the particle is circular.

### Step 6: Determine the direction of motion
Since \( x \) is defined as \( a \sin(\omega t + \delta) \) and \( y \) as \( a \cos(\omega t + \delta) \), the motion will be in a clockwise direction as \( t \) increases.

### Final Result
The two simple harmonic motions combine to produce a circular motion of radius \( a \) in the clockwise direction.

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