The time period T is defined as the smallest interval of time after which the motion repeats. Because the motion is periodic, it will repeat not just once, but indefinitely at intervals of T, 2T, 3T, and so on. Therefore, stating that it repeats 'only once' is physically incorrect as the oscillation continues.

In simple harmonic motion, the restoring force is given by Hooke's Law, F = -kx. According to Newton's second law, F = ma, so ma = -kx, which implies a = -(k/m)x. Since k and m are constants, the acceleration 'a' is directly proportional to the displacement 'x', but in the opposite direction.

A see-saw moves back and forth about a fixed pivot point. This repetitive to-and-fro motion about an equilibrium position is characteristic of vibratory or oscillatory motion. While it involves rotation, the specific back-and-forth nature is best described as vibratory.

The time period of a simple pendulum is given by T = 2π√(L/g_eff), where g_eff is the effective acceleration due to gravity. During free fall, the lift and the pendulum experience the same downward acceleration (g), resulting in an effective gravity of zero. As the denominator approaches zero, the time period mathematically approaches infinity, indicating the pendulum does not oscillate.

The restoring force in a simple pendulum is provided by the tangential component of gravity, which acts to return the bob to its equilibrium position. When the pendulum is displaced by an angle, gravity creates a torque or force that pulls it back toward the center. This is distinct from spring-mass systems where the restoring force is governed by Hooke's Law.

A seconds pendulum is defined as a pendulum with a period of exactly two seconds. This means it takes one second for a single swing in one direction and one second for the return swing, resulting in a frequency of 0.5 Hz. Note that the source text incorrectly states 12 Hz; the correct frequency is 0.5 Hz.

The natural frequency (f) of a simple pendulum is given by the formula f = (1/2π) * √(g/l), where g is the acceleration due to gravity and l is the length of the pendulum. Therefore, the frequency is proportional to the inverse of the square root of the length. The answer key identifies the square root relationship as the determining factor for the frequency calculation.

Simple harmonic motion is characterized by a restoring force that is directly proportional to the displacement, following Hooke's Law. When the displacement is plotted against time, the resulting graph is a sinusoidal function, specifically a sine or cosine wave. This smooth, periodic oscillation is the fundamental signature of SHM, representing the continuous conversion between potential and kinetic energy.

A seconds pendulum is defined as a pendulum that has a time period (T) of 2 seconds for one complete oscillation (a full back-and-forth cycle). Since frequency (f) is the reciprocal of the time period (f = 1/T), the frequency is 1/2, which equals 0.5 Hz.

A mass attached to a spring undergoes vibratory motion because it oscillates back and forth about a fixed equilibrium position. Furthermore, if the spring obeys Hooke's Law (F = -kx), the restoring force is directly proportional to the displacement, which is the defining characteristic of simple harmonic motion. Thus, it satisfies the criteria for both descriptions.