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A particule moves in x - y plane acordin...

A particule moves in x - y plane acording to rule x = a sin `omega t` and y = a cos `omega t`. The particle follows

A

a circular path

B

a parabolic path

C

a straight line path inclined equally to x and y-axes

D

an elliptical path

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the motion of the particle in the x-y plane given by the equations \( x = a \sin(\omega t) \) and \( y = a \cos(\omega t) \). ### Step-by-Step Solution: 1. **Write the equations**: The motion of the particle is described by the equations: \[ x = a \sin(\omega t) \] \[ y = a \cos(\omega t) \] 2. **Square both equations**: We square both equations to eliminate the trigonometric functions: \[ x^2 = a^2 \sin^2(\omega t) \] \[ y^2 = a^2 \cos^2(\omega t) \] 3. **Add the squared equations**: Now, we add these two equations: \[ x^2 + y^2 = a^2 \sin^2(\omega t) + a^2 \cos^2(\omega t) \] 4. **Factor out \( a^2 \)**: We can factor out \( a^2 \) from the right side: \[ x^2 + y^2 = a^2 (\sin^2(\omega t) + \cos^2(\omega t)) \] 5. **Use the Pythagorean identity**: We know from trigonometry that: \[ \sin^2(\omega t) + \cos^2(\omega t) = 1 \] Therefore, we can substitute this into our equation: \[ x^2 + y^2 = a^2 \cdot 1 \] \[ x^2 + y^2 = a^2 \] 6. **Identify the shape**: The equation \( x^2 + y^2 = a^2 \) represents a circle with radius \( a \) centered at the origin in the x-y plane. ### Conclusion: The particle follows a circular path. ### Final Answer: The correct option is (a) a circular path.
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