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Velocity in Simple Harmonic Motion
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Today, we will discuss the velocity of a particle in simple harmonic motion, or SHM. The displacement of a particle is given by the formula x(t) = A cos(Οt + Ο). Can anyone remind me how we find the velocity?
We differentiate the displacement with respect to time!
Exactly! So if we differentiate x(t), we get the velocity function. Can anyone help me write down the equation for velocity?
Itβs v(t) = -ΟA sin(Οt + Ο).
Very well. Notice that the velocity is sinusoidal as well. The maximum speed is given by ΟA. How does the negative sign affect our understanding of velocity?
It means that when the displacement is positive, the velocity is negative, showing it's moving towards the equilibrium position.
Great observation! This characteristic behavior leads to periodic motion. Remember: in SHM, the velocity is always changing direction. Let's summarize: velocity in SHM is derived from displacement and oscillates sinusoidally.
Acceleration in Simple Harmonic Motion
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Now let's dive into acceleration. Can anyone tell me how we derive the acceleration from the velocity function?
We differentiate the velocity function!
Exactly, and what do we get when we differentiate v(t)?
The acceleration function a(t) = -ΟΒ²A cos(Οt + Ο)!
Well done! Notice a key aspect: acceleration is proportional to displacement, but in the opposite direction. Can anyone explain why this is critical in SHM?
It shows that whenever the particle moves away from the equilibrium position, the acceleration brings it back, acting as a restoring force.
Exactly! This restoring behavior is what characterizes SHM. To wrap this up, the acceleration oscillates sinusoidally and is always directed toward the equilibrium position, just like velocity.
Connections between Displacement, Velocity, and Acceleration
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We have explored displacement, velocity, and acceleration in SHM. Let's discuss their interconnections. Who can explain the phase differences between these quantities?
Displacement reaches its maximum first, then the velocity reaches its maximum at the equilibrium position, and acceleration is at its maximum when displacement is at its maximum.
Spot on! The phase differences are very important: displacement leads velocity by 90 degrees, and velocity leads acceleration. This is crucial for understanding oscillatory motion. Can anyone summarize the relationships?
Displacement, velocity, and acceleration are all periodic functions. Each changes sinusoidally but has different maximum values and direction.
Exactly! So to recall: in SHM, velocity and acceleration are sinusoidal, and their phases affect how the system behaves. Excellent contributions today!
Introduction & Overview
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Quick Overview
Standard
In this section, we explore the instantaneous velocity and acceleration of a particle undergoing simple harmonic motion (SHM). The equations governing these quantities are derived from the displacement function, revealing their sinusoidal nature and their dependency on the amplitude and angular frequency of the motion.
Detailed
Velocity and Acceleration in Simple Harmonic Motion
In simple harmonic motion (SHM), the relationships between displacement, velocity, and acceleration are fundamental to understanding the nature of the motion. The displacement of a particle in SHM is expressed by the equation:
x(t) = A cos(Οt + Ο),
where:
- A is the amplitude
- Ο is the angular frequency
- Ο is the phase constant
Velocity in SHM
The velocity v(t) can be derived by differentiating the displacement with respect to time:
v(t) = -ΟA sin(Οt + Ο).
This equation indicates that:
- The velocity is a function of the sine of the angle, which varies between the maximum positive and negative values of Β±ΟA.
- The negative sign in the equation emphasizes the direction of velocity, indicating that it is opposite to the direction of displacement when the particle is at a positive position.
Acceleration in SHM
Similarly, the instantaneous acceleration a(t) can be calculated by differentiating the velocity function:
a(t) = -ΟΒ²A cos(Οt + Ο) = -ΟΒ²x(t).
Here, acceleration remains proportional to the displacement from the equilibrium position and is directed towards the center, denoting a restoring force.
Key Properties
- Phase Difference: There is a phase difference between the displacement, velocity, and acceleration in SHM. The displacement reaches maximum displacement while the velocity reaches maximum speed at the equilibrium position.
- Sinusoidal Variation: Both velocity and acceleration change sinusoidally over time, highlighting the harmonic nature of the motion. This relationship ensures that the motion is periodic and predictable.
- Direction of Acceleration: Acceleration acts in the opposite direction to the displacement, affirming its role as a restoring force.
Understanding these relationships is crucial for studying mechanical systems undergoing oscillatory motion, as they govern the behavior and characteristics of a variety of physical systems, including springs, pendulums, and more.
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Speed in Uniform Circular Motion
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The speed of a particle v in uniform circular motion is its angular speed Ο times the radius of the circle A.
v = Ο A (13.8)
Detailed Explanation
In uniform circular motion, a particle moves around a circular path at a constant speed. The speed (v) can be calculated using the formula where 'v' is equal to the angular speed (Ο) multiplied by the radius of the circle (A). Angular speed is a measure of how fast the particle is moving around the circle.
Examples & Analogies
Think of a car driving around a circular track at a steady speed. The further the car is from the center, the greater the distance it covers in the same amount of time, which is essentially what's described by the equation for speed in circular motion.
Instantaneous Velocity in SHM
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From the geometry of Fig. 13.11, it is clear that the velocity of the projection particle P' at time t is
v(t) = β ΟA sin (Οt + Ο) (13.9)
Detailed Explanation
When examining the projection of a particle moving in a circle (like our earlier example), we can determine its instantaneous velocity. The formula given shows that the velocity depends on the sine of the phase (Οt + Ο). The negative sign indicates that as the particle moves to the right in its oscillation, the velocity is treated as moving in the negative x-direction.
Examples & Analogies
Imagine a child on a swing. When the swing reaches the highest point on one side, the speed is zero because it pauses before swinging back. As it comes down and passes through the middle, itβs moving the fastest, which is similar to how the particles in SHM behave as they oscillate.
Calculating Instantaneous Velocity from Displacement
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We can, of course, obtain this equation without using geometrical argument, directly by differentiating (Eq. 13.4) with respect of t:
d( )dv(t) = x tt(13.10)
Detailed Explanation
This chunk reveals that we donβt need to rely purely on geometric understanding to derive the instantaneous velocity. By taking the derivative of the displacement function directly with respect to time (denoted as t), we can find the velocity function. This shows a fundamental concept in physics that relates to how calculus connects motion with change over time.
Examples & Analogies
Consider a car's speedometer, which tells you how fast you are going at a specific moment. Similar to how that speedometer measures the car's instantaneous speed by looking at the changes in position over time, we derive velocity from displacement as a function of time.
Acceleration in SHM
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The instantaneous acceleration of the projection particle P' is then
a(t) = β ΟΒ²A cos (Οt + Ο) = β ΟΒ²x(t) (13.11)
Detailed Explanation
Here, we see that acceleration is also described as a function of the cosine of the phase (Οt + Ο). Like velocity, acceleration also has a negative sign indicating that it is always directed towards the center of the oscillation, which is important for restoring the system back to equilibrium.
Examples & Analogies
Think of a rubber band. When you stretch it, you feel a force pulling it back toward the center. This restoring force is what causes the rubber band to oscillate back and forth, just like how acceleration in SHM always pulls the particle towards the mean position.
Phase Relationships in SHM
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For simplicity, let us put Ο = 0 and write the expression for x(t), v(t), and a(t):
x(t) = A cos Οt,
v(t) = βΟA sin Οt,
a(t) = βΟΒ²A cos Οt
Detailed Explanation
By assuming the phase constant Ο equals zero, we simplify our equations for displacement, velocity, and acceleration in simple harmonic motion. This helps to visualize the relationship among these variables, which fluctuate sinusoidally over time.
Examples & Analogies
If you were to watch a pendulum swinging back and forth on a clock without any initial movement to the left or right, you can see how displacement, speed, and the direction of the force (acceleration) all change smoothly and in a periodic way.
Phase Differences in SHM
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With respect to displacement plot, velocity plot has a phase difference of Ο/2 and acceleration plot has a phase difference of Ο.
Detailed Explanation
In SHM, the displacement, velocity, and acceleration don't peak at the same time. The velocity graph peaks a quarter cycle (or Ο/2 radians) ahead of displacement, and acceleration peaks half a cycle (or Ο radians) ahead of displacement. This phase difference is crucial in understanding the dynamics of the oscillating system.
Examples & Analogies
Picture a dancer performing a routine. The dancer moves in a flowing manner (displacement), then shifts into a leap (velocity), and finally lands softly (acceleration). The sequence and timing of their movements illustrate how these concepts relate but are not simultaneous.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Displacement in SHM: Expressed as x(t) = A cos(Οt + Ο)
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Velocity Function: Obtained by differentiation, v(t) = -ΟA sin(Οt + Ο)
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Acceleration Function: Obtained by further differentiation, a(t) = -ΟΒ²A cos(Οt + Ο)
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Periodic Nature of SHM: Velocity and acceleration are both sinusoidal and periodic functions.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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If a mass attached to a spring oscillates with an amplitude of 5 cm and an angular frequency of 2 rad/s, the maximum velocity it can achieve is 10 cm/s (ΟA).
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When displaced from rest, a pendulum's oscillatory motion can be modeled, demonstrating both velocity and acceleration's dependency on position as it swings.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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In SHM the pace is right, with x, v, a in sync at night!
π Fascinating Stories
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Imagine a child on a swing. At the highest point, they're still, but when they drop, they gain speed, only to slow as they return to the top.
π§ Other Memory Gems
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Dynamo for Displacement, Velocity, Acceleration - in SHM, they dance in a harmonic relation!
π― Super Acronyms
SIMPLE - Sinusoidal properties In Motion, Position Leads Energy
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Displacement
Definition:
The distance and direction of an object's change in position from the origin, typically represented as a function of time in oscillatory motion.
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Term: Velocity
Definition:
The rate of change of displacement with respect to time, which can be positive or negative depending on direction.
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Term: Acceleration
Definition:
The rate of change of velocity with respect to time, directed towards the equilibrium position in SHM.
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Term: Amplitude
Definition:
The maximum extent of a vibration or oscillation, measured from the position of equilibrium.
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Term: Angular Frequency
Definition:
The rate of oscillation expressed in radians per unit time, typically denoted by the symbol Ο.
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Term: Phase
Definition:
A specific point in the cycle of motion, often represented by the term (Οt + Ο).
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Term: Restoring Force
Definition:
The force that brings the particle back to its equilibrium position, typically represented as proportional to the negative displacement in SHM.