A particle undergoes two simple harmonic motions described by x_1 =...

The total displacement of the particle is given by the principle of superposition:

$x = x_1 + x_2$

Substituting $t = 0.0125$ into the equation for $x_1$:

$x_1 = 2.0 \sin(100 \pi \times 0.0125)$

$x_1 = 2.0 \sin(1.25 \pi) = 2.0 \sin \left(\dfrac{5\pi}{4}\right)<br><br>$x_1 = 2.0 \times \left(-\dfrac{1}{\sqrt{2}}\right) = -\sqrt{2} \approx -$1.414 \text{ cm}$

Substituting $t = 0.0125$ into the equation for $x_2$:

$x_2 = 2.0 \sin \left(120 \pi \times 0.0125 + \dfrac{\pi}{3}\right)$

x_2 = 2.0 \sin \left(1.5 \pi +\dfrac{\pi}{3}\right) =$2.0 \sin\left(\dfrac{3\pi}{2} + \dfrac{\pi}{3}\right)<br><br>Using the trigonometric identity$\sin\left(\dfrac{3\pi}{2} + \theta\right) = -\cos \theta:

$x_2 = -2.0 \cos\left(\dfrac{\pi}{3}\right) = -$2.0 \times $\dfrac{1}{2} = -$1.0 \text{ cm}$<br><br>The total displacement at$t = 0.0125$is:<br><br>$x = x_1 + x_2 = -1.414 - 1.0 = -2.414 \text{ cm}$<br><br>Rounding to two decimal places, the displacement is$-2.41 \text{ cm}$.$