A particle undergoes two simple harmonic motions, one along the X-axis and the other along a line at an angle of $45^{}^{{}^{\circ}}$ to the X-axis. These motions are described by
$x = x_0 \sin\omega t$ and $s = s_0 \sin\omega t$ .
Determine the amplitude of the resultant motion.

Question

A particle undergoes two simple harmonic motions, one along the X-axis and the other along a line at an angle of

45^{}^{{}^{\circ}}

to the X-axis. These motions are described by

x = x_0 \sin\omega t

and

s = s_0 \sin\omega t

.
Determine the amplitude of the resultant motion.

Quantrex Academy · Accepted Answer

Correct answer: B. [ x_0^2 + s_0^2 + 2 \, x_0 s_0 ]^ 1/2

The two simple harmonic motions are given by

\vec{x} = x_0 \sin\omega t \, \hat{i}

and

\vec{s} = s_0 \sin\omega t (\cos 45^{}^{{}^{\circ}} \hat{i} + \sin 45^{}^{{}^{\circ}} \hat{j})

.

Since both simple harmonic motions are in the same phase, the resultant amplitude is the vector sum of their individual amplitudes.

The angle between the two amplitude vectors is

\theta = 45^{}^{{}^{\circ}}

.

Using the vector addition formula, the resultant amplitude

A

is given by:

A = \sqrt{x_0^2 + s_0^2 + 2 x_0 s_0 \cos 45^{}^{{}^{\circ}}}

A = \sqrt{x_0^2 + s_0^2 + 2 x_0 s_0 \left(\dfrac{1}{\sqrt{2}}\right)}

A = \left[ x_0^2 + s_0^2 + \sqrt{2} x_0 s_0 \right]^{1/2}

A particle undergoes two simple harmonic motions, one along the...

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