### Understanding the Oscillation Amplitude of a Damped Harmonic Oscillator**Concept Overview**: A lightly damped harmonic oscillator, with a natural (undamped) angular frequency \( \omega_0 \), is being driven by an external force at a different angular frequency \( \omega_d \). The problem is to determine which of the provided traces in the graph best represents the relationship between driving frequency \( \omega_d / \omega_0 \) and oscillation amplitude \( A / A_0 \).**Graph Analysis**:- **Axes**: - **X-axis**: Represents the normalized driving angular frequency, \( \omega_d / \omega_0 \). - **Y-axis**: Represents the normalized amplitude, \( A / A_0 \).- **Graph Traces**: - **A1 (Purple)**: Shows a peak around \( \omega_d / \omega_0 \approx 0.5 \) and quickly drops towards zero as \( \omega_d / \omega_0 \) increases. - **A2 (Green, dashed)**: Peaks sharply around \( \omega_d / \omega_0 \approx 1.5 \) and then declines. - **A3 (Cyan, dash-dot)**: Displays a wider peak around \( \omega_d / \omega_0 \approx 1 \) and decreases as \( \omega_d / \omega_0 \) increases. - **A4 (Orange, solid)**: Low and flat, with a minimal peak near \( \omega_d / \omega_0 \approx 0.5 \). - **A5 (Blue, dotted)**: Peaks notably around \( \omega_d / \omega_0 \approx 0.75 \), tapering off thereafter.**Question**: Which trace best matches the function of amplitude \( A(\omega_d / \omega_0) \) for this lightly damped oscillator? The solution would involve understanding resonance peaks typical in such oscillatory systems and selecting the appropriate trace based on the behavior described.**Answer Choices**:- O A1- O A2- O A3- O A4- O A5Use this graph to explore concepts such as resonant frequency and damping in oscillatory systems. The identified pattern helps in visualizing how amplitude varies with changing driving frequencies in practical applications.

Name: ### Understanding the Oscillation Amplitude of a Damped Harmonic Osci
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Description: ### Understanding the Oscillation Amplitude of a Damped Harmonic Oscillator**Concept Overview**: A lightly damped harmonic oscillator, with a natural (undamped) angular frequency \( \omega_0 \), is being driven by an external force at a different a

When the angular frequency is closest to (equal to) the driving frequency, the system reaches the resonance state. At this stage, it oscillates at the highest amplitude.So, when the angular frequency () is equal to the driving frequency (), the amplitude will be the highest. This characteristic can be seen in the plot of A3 because it has the highest amplitude.A3 is closest to A () behaviour of the oscillation amplitude vs the driving frequency.

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A lightly damped harmonic oscillator (natural, undamped angular frequency w o) is driven at an angular frequency w d. Which of the traces in the graph below is closest to the A(w d/ wo) behavior of its oscillation amplitude vs. driving frequency (in units of wo)? Amplitude A/A 00000 A1 OA2 A3 A4 O A5 10 8 6 4 2 O 0 A1 A2 A3 A4 A5.... 0.5 1 1.5 2 2.5 Normalized Driving Angular Frequency wa/wo 3

A lightly damped harmonic oscillator (natural, undamped angular frequency w o) is driven at an angular frequency w d. Which of the traces in the graph below is closest to the A(w d/ wo) behavior of its oscillation amplitude vs. driving frequency (in units of wo)? Amplitude A/A 00000 A1 OA2 A3 A4 O A5 10 8 6 4 2 O 0 A1 A2 A3 A4 A5.... 0.5 1 1.5 2 2.5 Normalized Driving Angular Frequency wa/wo 3

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Concept explainers

Simple harmonic motion

Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.

Simple Pendulum

A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.

Oscillation

In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.

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Question

### Understanding the Oscillation Amplitude of a Damped Harmonic Oscillator

**Concept Overview**:
A lightly damped harmonic oscillator, with a natural (undamped) angular frequency \( \omega_0 \), is being driven by an external force at a different angular frequency \( \omega_d \). The problem is to determine which of the provided traces in the graph best represents the relationship between driving frequency \( \omega_d / \omega_0 \) and oscillation amplitude \( A / A_0 \).

**Graph Analysis**:

- **Axes**:
- **X-axis**: Represents the normalized driving angular frequency, \( \omega_d / \omega_0 \).
- **Y-axis**: Represents the normalized amplitude, \( A / A_0 \).

- **Graph Traces**:
- **A1 (Purple)**: Shows a peak around \( \omega_d / \omega_0 \approx 0.5 \) and quickly drops towards zero as \( \omega_d / \omega_0 \) increases.
- **A2 (Green, dashed)**: Peaks sharply around \( \omega_d / \omega_0 \approx 1.5 \) and then declines.
- **A3 (Cyan, dash-dot)**: Displays a wider peak around \( \omega_d / \omega_0 \approx 1 \) and decreases as \( \omega_d / \omega_0 \) increases.
- **A4 (Orange, solid)**: Low and flat, with a minimal peak near \( \omega_d / \omega_0 \approx 0.5 \).
- **A5 (Blue, dotted)**: Peaks notably around \( \omega_d / \omega_0 \approx 0.75 \), tapering off thereafter.

**Question**:
Which trace best matches the function of amplitude \( A(\omega_d / \omega_0) \) for this lightly damped oscillator? The solution would involve understanding resonance peaks typical in such oscillatory systems and selecting the appropriate trace based on the behavior described.

**Answer Choices**:
- O A1
- O A2
- O A3
- O A4
- O A5

Use this graph to explore concepts such as resonant frequency and damping in oscillatory systems. The identified pattern helps in visualizing how amplitude varies with changing driving frequencies in practical applications.

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