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The two-dimensional motion of a particle, described by $\vec{r}=(\hat{i}+2\hat{j})A\cos\omega t$ is a/an: A. parabolic path B. elliptical path C. periodic motion D. 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
Detailed Solution
Since $x=A\cos\omega t$ and $y=2A\cos\omega t$, we get $y=2x$, a straight line — not parabolic or elliptical. $\dfrac{d\vec r}{dt}=-(\hat i+2\hat j)\omega A\sin\omega t$ and $\dfrac{d^2\vec r}{dt^2}=-\omega^2\vec r$, i.e. $\vec a=-\omega^2\vec r$, which is the defining condition of SHM, and SHM is periodic. So only C and D apply.