13.2.2. Displacement

In section 3.2, we defined displacement of a particle as the change in its position vector. In this chapter, we use the term displacement in a more general sense. It refers to change with time of any physical property under consideration.

Displacement generally refers to a change from an initial position to a final position over time. In physics, it's often used only to describe the change in position of an object, but here we broaden this definition to include any physical property that varies over time. This means displacement can refer to changes in position, voltage in electrical circuits, pressure changes in sound waves, or variations in the electric and magnetic fields in light waves.

Imagine a rubber band being stretched. The change in its length as you stretch it and release it can be thought of as a form of displacement, not just in terms of position, but in how the rubber band changes shape and tension over time.

For example, in case of rectilinear motion of a steel ball on a surface, the distance from the starting point as a function of time is its position displacement. The choice of origin is a matter of convenience.

In any physical situation, we can choose our starting point, or origin, based on what makes calculations easier. For instance, if we are tracking the movement of a steel ball along a straight path, we might consider the initial position of the ball as zero (the origin). As time progresses, the displacement is measured as the distance from this starting point, indicating how far the ball has moved.

Think of a race track: if you define the starting line as your zero mark, you can easily calculate how many meters a runner has traveled throughout the race from that point.

For an oscillating simple pendulum, the angle from the vertical as a function of time may be regarded as a displacement variable. The term displacement is not always to be referred in the context of position only.

In systems that oscillate, such as a pendulum, displacement is not just about linear movement along a path. Instead, it can refer to angular changes as well. For example, when a pendulum swings, its displacement can be measured as the angle it makes from the vertical position. This highlights that displacement can also include angles and various other parameters, broadening our understanding of movement.

Consider a clock's pendulum: as it swings left and right, it makes a specific angle with the vertical. Watching this swinging motion, you can think of the angle as a form of displacement that defines its position in the swing rather than linear distance.

There can be many other kinds of displacement variables. The voltage across a capacitor, changing with time in an AC circuit, is also a displacement variable. In the same way, pressure variations in time in the propagation of sound wave, the changing electric and magnetic fields in a light wave are examples of displacement in different contexts.

Displacement can apply to various physical phenomena beyond just position measurements. In electricity, for instance, the changing voltage across a capacitor can be considered a form of displacement because it indicates how electrical properties vary over time. Similarly, sound waves involve displacements in pressure, allowing the wave to travel through the medium. In light waves, displacement could refer to the changing electric and magnetic field strengths as they propagate.

Think of sound as a wave traveling through water: as a stone is thrown into a pond, it creates ripples that displace the water surface at each point. Similarly, as sound travels, it displaces the air particles around it, creating variations in pressure that allow us to hear.

The displacement can be represented by a mathematical function of time. In case of periodic motion, this function is periodic in time.

In mathematics, we can express displacement using functions to describe how it changes over time. For periodic motion, there's often a predictable pattern that repeats. This means we can use functions like sine or cosine to represent the displacement of an object over time, where the periodic nature reflects the regular repetition of motion.

Imagine someone on a swing: their position as a function of time can be graphed, and you would see a repeating wave-like pattern that shows how they swing back and forth. This can be precisely represented with a mathematical function.

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