13.2.2 - Displacement
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Understanding Displacement
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Today, we will discuss displacement in the context of oscillatory motion. Displacement refers to any change in position or physical property over time. For instance, if a ball rolls down a ramp, we measure its displacement as the distance it moves from the starting point.
So, displacement is not just about position, is it?
Exactly! Displacement can refer to changes in voltage, pressure, or any other property over time. It's a broader term that encompasses various contexts.
Can you give an example of displacement in AC circuits?
In AC circuits, the voltage varies over time. You can think of the voltage as a displacement variable that oscillates around an average value.
Interesting! So could this also apply to sound waves?
Yes, absolutely! Sound waves involve pressure variations, which can also be interpreted as displacement in a different context.
How is displacement mathematically represented?
Good question! Displacement can be expressed as a periodic function, like this: $$ f(t) = A \cos(\omega t) $$ where $$ A $$ is the amplitude, and $$ \omega $$ is the angular frequency.
To sum up, displacement can refer to various physical properties, not just position. It's a critical concept in understanding oscillations and waves.
Periodic Functions and Displacement
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Now let's dive deeper into periodic functions and how they relate to displacement. Periodic functions repeat their values in regular intervals, which is essential in oscillatory motions.
How do we identify a periodic function?
A function is periodic if it can be described by an equation like $$ f(t) = A \cos(\omega t) $$ and retains the same value at regular time intervals—this is crucial in any oscillatory system.
How do we find the period of such functions?
The period is found using the equation $$ T = \frac{2\pi}{\omega} $$, which indicates how long it takes for the displacement function to repeat itself.
Are there other forms of this function we should know?
Certainly! A linear combination of sine and cosine functions is also periodic. For example, $$ f(t) = A \sin(\omega t) + B \cos(\omega t) $$ still represents a periodic motion.
So, basically, all these forms help us understand different types of oscillatory systems?
Exactly! Understanding these relationships is vital in physics to describe many physical phenomena, including sound and electromagnetic waves.
Introduction & Overview
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Quick Overview
Standard
In this section, displacement is defined as a general term for change in any physical property over time, illustrated through examples in rectilinear motion, oscillations, and AC circuits. The mathematical representation of displacement as a periodic function is also discussed.
Detailed
Detailed Summary
In this section on Displacement, we explore the broader implications of displacement within oscillatory motion. Displacement is not merely a positional change; it encompasses variances in other physical properties such as voltage and pressure, particularly in oscillatory systems.
Examples of displacement include:
- Rectilinear Motion: For a steel ball rolling on a surface, the distance from its starting point represents its displacement as a function of time.
- Oscillatory Systems: Consider a mass on a spring oscillating around its equilibrium position, where its position varies from that midpoint.
- Physical Properties: Displacement can manifest in the voltage across capacitors in AC circuits or variations in pressure due to sound waves.
Mathematically, displacement is expressed as a periodic function of time, such as:
$$ f(t) = A \cos(\omega t) $$
where
- $$ A $$ represents the amplitude,
- $$ \omega $$ is the angular frequency,
- The argument $$ \omega t $$ indicates periodicity as it cycles through values, reinforcing the relationship between displacement and time in periodic motion.
The period is deduced from the function, emphasizing the repetitive nature of oscillations.
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Definition of Displacement
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Chapter Content
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.
Detailed Explanation
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.
Examples & Analogies
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.
Contextual Use of Displacement
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Displacement in Oscillating Systems
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Displacement Variables Beyond Position
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Mathematical Representation of Displacement
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The displacement can be represented by a mathematical function of time. In case of periodic motion, this function is periodic in time.
Detailed Explanation
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.
Examples & Analogies
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.
Key Concepts
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Displacement: Refers to any change in position or physical property with time.
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Periodic Motion: A repetitive motion that occurs at regular time intervals.
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Amplitude: The peak value of the oscillation or displacement from the mean position.
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Angular Frequency: A measure of how rapidly something oscillates, denoted as $$ \omega $$.
Examples & Applications
A mass attached to a spring oscillates around its equilibrium position, and its displacement varies accordingly.
Voltage in an AC circuit changes periodically, providing an example of displacement in electrical systems.
Memory Aids
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Rhymes
Displacement's the change we can gauge, from motion to voltage, it's all on the stage!
Stories
Imagine a yo-yo, going up and down. The string changes length, that's displacement's crown! Voltage and pressure, they too have their part, showing us movement—displacement is smart.
Memory Tools
DPAV: Displacement, Periodic, Amplitude, Velocity - remembering the key concepts.
Acronyms
DPA
Displacement
Periodicity
Amplitude!
Flash Cards
Glossary
- Displacement
The change in position or any physical property over time.
- Periodic Motion
Motion that repeats itself at regular intervals.
- Amplitude
The maximum extent of a periodic variable.
- Angular Frequency
The rate of change of the phase of a sinusoidal waveform, typically represented by $$ \omega $$.
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