C.1 Simple harmonic motion

SHM is defined by the equation $a=-{\omega }^{2}x$, which tells us the acceleration (and hence restoring force) is directly proportional to displacement $x$.

The negative sign in $a=-{\omega }^{2}x$ means acceleration always opposes displacement — i.e., the restoring force is always directed toward the equilibrium position.

Both conditions are: (1) restoring force proportional to displacement, and (2) restoring force directed toward equilibrium. This matches the second option.

The question asks for both defining conditions of SHM simultaneously.

"The restoring force is constant in magnitude and always directed away from equilibrium" is wrong on both counts — SHM requires the force to vary with displacement and point toward equilibrium.

"Proportional to the square of displacement" does not satisfy SHM; $F\propto x^2$ produces oscillatory motion but not simple harmonic motion.

"Proportional to velocity" describes a damping force, not a restoring force. This would actually reduce oscillation amplitude over time.

The remaining option — proportional to displacement, directed toward equilibrium — correctly states both SHM conditions.

Angular frequency is related to period by: $\omega =\frac{2\pi }{T}$

$\omega =\frac{2\pi }{0.25}=8\pi \approx 25.1{\text{ rad s}}^{-1}$

The exact answer is $8\pi$ rad s⁻¹, which is also approximately $25.1$ rad s⁻¹. Since $8\pi$ is the exact form and matches an option, that is the best answer.

We need $\omega$ given $T=0.25$ s using $\omega =2\pi /T$.

$4.0=1/0.25$, which is simply $1/T=f$, the frequency in Hz — not the angular frequency.

$\pi /2\approx 1.57$ rad s⁻¹ is far too small for a period of $0.25$ s. This would correspond to a period of about $4$ s.

$25.1\approx 8\pi$, so numerically it is correct. However, $8\pi$ is the exact form and is preferable in IB answers. Both represent the same value, but $8\pi$ is listed as the distinct correct option.

The correct answer is $8\pi$ rad s⁻¹, which equals $\frac{2\pi }{0.25}$.