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, we need to analyze the two simple harmonic motions given by the equations:

1. \( x = a \sin(\omega t + \delta) \)
2. \( y = a \sin(\omega t + \delta + \frac{\pi}{2}) \)

### Step 1: Rewrite the second equation
The second equation can be rewritten using the sine addition formula. We know that:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
Using this, we can express \( y \) as:
\[ y = a \sin(\omega t + \delta + \frac{\pi}{2}) = a \left( \sin(\omega t + \delta) \cos\left(\frac{\pi}{2}\right) + \cos(\omega t + \delta) \sin\left(\frac{\pi}{2}\right) \right) \]
Since \(\cos\left(\frac{\pi}{2}\right) = 0\) and \(\sin\left(\frac{\pi}{2}\right) = 1\), we have:
\[ y = a \cos(\omega t + \delta) \]

### Step 2: Substitute the equations
Now we have:
1. \( x = a \sin(\omega t + \delta) \)
2. \( y = a \cos(\omega t + \delta) \)

### Step 3: Find the relationship between \(x\) and \(y\)
Next, we can square both equations and add them:
\[ x^2 = a^2 \sin^2(\omega t + \delta) \]
\[ y^2 = a^2 \cos^2(\omega t + \delta) \]
Adding these two equations gives:
\[ x^2 + y^2 = a^2 (\sin^2(\omega t + \delta) + \cos^2(\omega t + \delta)) \]
Using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\):
\[ x^2 + y^2 = a^2 \]

### Step 4: Identify the type of motion
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 5: Determine the direction of motion
To determine the direction of motion, we can analyze the phase of the sine and cosine functions. As \(t\) increases, \(\sin(\omega t + \delta)\) increases from 0 to 1, while \(\cos(\omega t + \delta)\) decreases from 1 to 0. This indicates that the motion is in the clockwise direction.

### Final Answer
The two simple harmonic motions act on the particle in a **circular clockwise** motion.

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