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Ch 14: Periodic Motion
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 14: Periodic MotionProblem 4cd
Chapter 14, Problem 4cd

The displacement of an oscillating object as a function of time is shown in Fig. E14.4. What is (c) the period? (d) the angular frequency of this motion?
Graph depicting the displacement of an oscillating object as a function of time in simple harmonic motion.

Verified step by step guidance
1
To find the amplitude, observe the maximum displacement from the equilibrium position on the graph. The amplitude is the absolute value of this maximum displacement.
To determine the period, identify the time it takes for the oscillating object to complete one full cycle. This can be found by measuring the time interval between two consecutive points where the object is at the same position and moving in the same direction.
The frequency is the reciprocal of the period. Use the formula: \( f = \frac{1}{T} \), where \( f \) is the frequency and \( T \) is the period.
The angular frequency can be calculated using the formula: \( \omega = 2\pi f \), where \( \omega \) is the angular frequency and \( f \) is the frequency.
Review the graph to ensure that the measurements for amplitude, period, and frequency are consistent with the visual data, and adjust if necessary.

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Key Concepts

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

Period

The period of an oscillating object is the time it takes to complete one full cycle of motion. In the context of simple harmonic motion, it is the duration between successive peaks or troughs in the displacement graph. From the provided graph, the period can be determined by measuring the time interval between two consecutive maximum or minimum points.
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Angular Frequency

Angular frequency, denoted by the symbol ω, is a measure of how quickly an object oscillates in radians per second. It is related to the period (T) by the formula ω = 2π/T. This concept is crucial for understanding the dynamics of oscillatory motion, as it provides insight into the speed of oscillation in relation to the circular motion analogy.
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Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement from an equilibrium position and acts in the opposite direction. This motion can be represented graphically as a sine or cosine wave, as seen in the displacement vs. time graph. Understanding SHM is essential for analyzing the behavior of oscillating systems, such as springs and pendulums.
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Related Practice
Textbook Question

In a physics lab, you attach a 0.200-kg air-track glider to the end of an ideal spring of negligible mass and start it oscillating. The elapsed time from when the glider first moves through the equilibrium point to the second time it moves through that point is 2.60 s. Find the spring's force constant.

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Textbook Question

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7 shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

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Textbook Question

Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern that has the same frequency as the note that is sung. If someone sings the note B flat, which has a frequency of 466 Hz, how much time does it take the person's vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords?

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Textbook Question

A machine part is undergoing SHM with a frequency of 4.00 Hz and amplitude 1.80 cm. How long does it take the part to go from x = 0 to x = -1.80 cm ?

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Textbook Question

The displacement of an oscillating object as a function of time is shown in Fig. E14.4. What is (a) the frequency? (b) the amplitude?

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