State whether the following statements are True or False:
$1.$ If a spring is cut into two equal pieces,the spring constant of each piece decreases.
$2.$ As the displacement of a Simple Harmonic Oscillator $(SHO)$ increases,its acceleration decreases.
$3.$ $A$ system can oscillate with more than one natural frequency.
$4.$ The periodic time of Simple Harmonic Motion $(SHM)$ depends on amplitude,energy,or phase constant.

(A-D) $1.$ False. When a spring is cut into two equal pieces,the spring constant of each piece doubles $(k' = 2k)$.
$2.$ False. In $SHO$,acceleration $a = -\omega^2 x$. As the magnitude of displacement $|x|$ increases,the magnitude of acceleration $|a|$ increases.
$3.$ True. Complex systems (like coupled oscillators or vibrating strings) can have multiple natural frequencies (normal modes).
$4.$ False. The periodic time $T = 2\pi \sqrt{m/k}$ of an ideal $SHM$ is independent of amplitude,energy,and phase constant.