Question

The two-dimensional motion of a particle, described by \( \vec{r} = (\hat{i} + 2\hat{j}) A \cos \omega t \) is a/an:
1. parabolic path
2. elliptical path
3. periodic motion
4. simple harmonic motion
Choose the correct answer from the options given below:

• A. B, C and D only
• B. A, B and C only
• C. A, C and D only
• D. C and D only

Answer 1

The Correct Option is D

Let the position vector be \(\vec{r} = x\hat{i} + y\hat{j}\). Then:

x = A cos ωt

y = 2A cos ωt

This represents simple harmonic motion along both the x and y axes.

$\frac{x}{A}$ = cos ωt

$\frac{y}{2A}$ = cos ωt

Therefore: $\frac{x}{A} = \frac{y}{2A}$

y = 2x

This is the equation of a straight line, so the path is not parabolic or elliptical.

The motion is periodic and simple harmonic along the line y = 2x.