Ch 15: Oscillations

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 15: Oscillations

Problem 73

Chapter 15, Problem 73

A 200 g oscillator in a vacuum chamber has a frequency of 2.0 Hz. When air is admitted, the oscillation decreases to 60% of its initial amplitude in 50 s. How many oscillations will have been completed when the amplitude is 30% of its initial value?

Verified step by step guidance

Step 1: Identify the key parameters of the problem. The mass of the oscillator is 200 g (convert to kilograms: 0.2 kg), the natural frequency is 2.0 Hz, the initial amplitude decreases to 60% in 50 s, and we need to find the number of oscillations when the amplitude is 30% of its initial value.

Step 2: Use the concept of damped harmonic motion. The amplitude of a damped oscillator decreases exponentially over time, following the equation: \( A(t) = A_0 e^{-b t / 2m} \), where \( A_0 \) is the initial amplitude, \( b \) is the damping coefficient, \( m \) is the mass, and \( t \) is time.

Step 3: Solve for the damping coefficient \( b \). From the problem, when \( t = 50 \) s, \( A(t) = 0.6 A_0 \). Substitute these values into the amplitude equation: \( 0.6 A_0 = A_0 e^{-b (50) / (2 \cdot 0.2)} \). Simplify to find \( b \): \( 0.6 = e^{-b (50) / 0.4} \). Take the natural logarithm of both sides to solve for \( b \).

Step 4: Determine the time \( t \) when the amplitude is 30% of the initial value. Use the same amplitude equation: \( 0.3 A_0 = A_0 e^{-b t / 2m} \). Simplify to \( 0.3 = e^{-b t / 0.4} \). Take the natural logarithm of both sides and solve for \( t \) using the value of \( b \) found in Step 3.

Step 5: Calculate the number of oscillations completed by the oscillator in the time \( t \). The number of oscillations is given by \( N = f \cdot t \), where \( f \) is the frequency (2.0 Hz). Multiply the frequency by the time \( t \) to find the total number of oscillations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Damping

Damping refers to the reduction in amplitude of oscillations over time due to energy loss, often caused by friction or resistance. In this scenario, the introduction of air creates a damping effect on the oscillator, leading to a decrease in amplitude as energy is dissipated. Understanding damping is crucial for analyzing how oscillatory systems behave in different environments.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of oscillations, the amplitude of the oscillator decreases exponentially over time due to damping. This concept is essential for calculating how long it takes for the amplitude to reach a certain percentage of its initial value.

Frequency and Period of Oscillation

Frequency is the number of oscillations per unit time, while the period is the time taken for one complete oscillation. In this problem, the oscillator has a frequency of 2.0 Hz, meaning it completes 2 oscillations every second. Understanding the relationship between frequency and period is vital for determining the total number of oscillations completed over a given time interval.

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