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:
1. parabolic path
2. elliptical path
3. periodic motion
4. simple harmonic motion
Choose the correct answer from the options given below:
- B, C and D only
- A, B and C only
- A, C and D only
- C and D only
The Correct Option is D
Solution and Explanation
Given the position vector \(\vec{r} = x\hat{i} + y\hat{j}\), where:
x = A cos ωt
y = 2A cos ωt
This indicates simple harmonic motion along both the x and y axes.
From these equations, we derive:
$\frac{x}{A}$ = cos ωt
$\frac{y}{2A}$ = cos ωt
Equating these gives: $\frac{x}{A} = \frac{y}{2A}$
Which simplifies to: y = 2x
This equation represents a straight line, thus ruling out parabolic or elliptical paths.
The motion is periodic and simple harmonic, occurring along the line defined by y = 2x.
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