13.9. SUMMARY

1. The motion that repeats itself is called periodic motion.

2. The period T is the time required for one complete oscillation, or cycle. It is related to the frequency ν by,

   ν = 1/T. The frequency ν of periodic or oscillatory motion is the number of oscillations per unit time. In the SI, it is measured in hertz :

   1 hertz = 1 Hz = 1 oscillation per second = 1 s⁻¹

Periodic motion is one that repeats consistently over time. The period (T) is the duration needed to complete one full cycle of the motion. The frequency (ν) indicates how often this cycle occurs per second. They are mathematically linked: if you know T, you can calculate ν as the reciprocal (1/T). For instance, if a swing takes 2 seconds to complete one forward and backward motion, its frequency is 0.5 Hz.

Think of a pendulum in a clock that swings back and forth. If it takes 2 seconds for it to return to the same position, it has a period of 2 seconds. Therefore, it ticks twice every 4 seconds, giving it a frequency of 0.5 ticks per second.

3. In simple harmonic motion (SHM), the displacement x(t) of a particle from its equilibrium position is given by,

   x(t) = A cos(ωt + φ) (displacement),

   in which A is the amplitude of the displacement, the quantity (ωt + φ) is the phase of the motion, and φ is the phase constant. The angular frequency ω is related to the period and frequency of the motion by,

   ω = 2π/T = 2πν (angular frequency).

Simple Harmonic Motion (SHM) describes a type of oscillation where the displacement of a particle is a cosine function of time. The amplitude (A) indicates the maximum displacement from the resting position. The angular frequency (ω) relates to how fast the oscillation occurs and connects to the period (T) and frequency (ν), telling you how many cycles happen within a given time frame.

Imagine a swing at a playground. If you pull the swing to one side (that’s the amplitude), and let go, it moves back and forth with a regular pattern. The motion can be predicted using this cosine function where time and displacement take part in the predictable rhythm of the swing.

4. Simple harmonic motion can also be viewed as the projection of uniform circular motion on the diameter of the circle in which the latter motion occurs.

This concept illustrates that simple harmonic motion can be understood as a shadow or projection created by a point on the circumference of a circle as it rotates. The horizontal movement of this point corresponds to the to-and-fro motion experienced in SHM. Both motions share the same mathematical behavior despite appearing different—one being circular and the other linear.

Picture a Ferris wheel. As one of the seats moves around the circle, if you were to view it from the side, the height of the seat would go up and down in a smooth, predictable manner like SHM. The vertical distance changes but the base movement is circular—this shows how two motion types are actually linked.

5. The particle velocity and acceleration during SHM as functions of time are given by,

   v(t) = –ωA sin(ωt + φ) (velocity), a(t) = –ω²A cos(ωt + φ) = –ω²x(t) (acceleration).

In SHM, as the particle moves back and forth, its velocity changes direction; it's maximum when passing the equilibrium point and zero at the extremes. The acceleration is similarly derived from the property of SHM, indicating it is always directed towards the mean position (rest point), which ensures that the motion oscillates around this central point.

Think about riding a skateboard down a ramp. At the bottom, you’re moving fastest (maximum velocity), and as you climb up the other side, your speed decreases until you stop momentarily (zero velocity) at the top before rolling back down. This shows both velocity and acceleration changing in an oscillatory fashion.

6. The force acting in a simple harmonic motion is proportional to the displacement and is always directed towards the centre of motion.
7. A particle executing simple harmonic motion has, at any time, kinetic energy K = ½mv² and potential energy U = ½kx². If no friction is present the mechanical energy of the system, E = K + U always remains constant even though K and U change with time.

In SHM, the force restoring the particle to its mean position is directly proportional to how far it is displaced. The kinetic energy (K) is highest when the particle is at the equilibrium position and lowest at the extremes, where potential energy (U) is maximum. The total mechanical energy (E) combines both types of energy and remains constant in the absence of external forces, reflecting energy transfer between kinetic and potential forms during oscillations.

Imagine a playground swing. When the swing is at its highest points, it has maximum potential energy, and as it swings down to the lowest point, all that energy is converted to kinetic energy. This constant switch between forms of energy illustrates conservation in SHM: even as one increases, the other decreases, resulting in a constant total energy.

8. The motion of a simple pendulum swinging through small angles is approximately simple harmonic. The period of oscillation is given by,

   T = 2π√(L/g).

A simple pendulum executes oscillatory motion when displaced slightly from its vertical resting position. For small angles, the motion approaches SHM, where the period (T) depends only on the length of the pendulum (L) and the acceleration due to gravity (g). This relationship shows that longer pendulums take more time to complete a swing.

Think about going to a clock tower with a swinging pendulum. If the pendulum is longer, it takes a longer time to return to its original position compared to a shorter pendulum, similar to how longer swings take longer to swing back and forth.

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