Ncert Exercises & Solutions

A particle is acted simultaneously by mutually perpendicular simple harmonic motion $x=a \text{cos}𝜔𝑡$ and $y = a\text{sin} 𝜔 𝑡$ . The trajectory of motion of the particle will be:

1. an ellipse

2. a parabola

3. a circle

4. a straight line

(3) Hint: The equation of motion gives the type of motion.

Step 1: Find an equation between x and y.

Given, x= a cos

ω t

$ω t$ ...(i)

y= asin

ω t

$ω t$ ...(ii)

Squaring and adding Eqs. (i) and (ii):

x^{2} + y^{2} = a^{2} ({cos}^{2} ω t + {sin}^{2} ω t) = a^{2}

$x^{2} + y^{2} = a^{2} (\cos^{2} ω t + \sin^{2} ω t) = a^{2}$

$\Rightarrow x^{2} + y^{2} = a^{2} [∵ \cos^{2} ω t + \sin^{2} ω t = 1]$

Step 2: Find the type of motion.

This is the equation of a circle.

Clearly, the locus is a circle of constant radius a.

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