Representing and Analyzing SHM Practice Test

A frictionless mass–spring oscillator has $m=0.10\ \text{kg}$, $k=40\ \text{N/m}$, and equilibrium at $x=0$. It is released from rest at $x=+0.050\ \text{m}$ at $t=0$, so $A=0.050\ \text{m}$ and $\phi=0$. The motion is $x(t)=A\cos(\omega t)$ with $\omega=\sqrt{k/m}$ and period $T=2\pi/\omega$. The object reaches equilibrium after one quarter period. All quantities use SI units and amplitude remains constant. How long after release does it first reach $x=0$?

A 0.30 kg mass on a spring with $k=120\ \text{N/m}$ oscillates without friction about equilibrium $x=0$. At $t=0$, the mass passes through equilibrium moving in the positive direction with speed $0.60\ \text{m/s}$. Using $x(t)=A\cos(\omega t+\phi)$ and $v(t)=-A\omega\sin(\omega t+\phi)$ with $\omega=\sqrt{k/m}$, the condition $x(0)=0$ implies $\cos\phi=0$. The sign of $v(0)$ sets the phase quadrant. All values are SI units and amplitude is constant. What is the phase constant $\phi$?