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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 7
Chapter 14, Problem 7

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?
Graph depicting the oscillation of a ball on a spring, showing position over time.

Verified step by step guidance
1
To find the period (T) of the oscillation, observe the time it takes for the ball to complete one full cycle on the graph. From the second image, the ball completes one cycle from 0 to 1 second, so T = 1.0 s.
The frequency (f) is the reciprocal of the period. Use the formula: f = 1/T. Substitute the period you found in the previous step to calculate the frequency.
The angular frequency (ω) is related to the frequency by the formula: ω = 2πf. Use the frequency calculated in the previous step to find the angular frequency.
The amplitude (A) is the maximum displacement from the equilibrium position. From the graph, the maximum displacement is 3.0 cm, so A = 3.0 cm.
The force constant (k) of the spring can be found using the formula: k = mω², where m is the mass of the ball (2.40 kg) and ω is the angular frequency. Substitute the values to find the force constant.

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Key Concepts

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

Oscillation

Oscillation refers to the repetitive variation, typically in time, of some measure about a central value. In the context of a mass-spring system, it describes how the mass moves back and forth around an equilibrium position due to the restoring force of the spring. The motion is periodic, characterized by parameters such as period, frequency, and amplitude.
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Period and Frequency

The period (T) of an oscillation is the time taken for one complete cycle of motion, while frequency (f) is the number of cycles per unit time, typically measured in hertz (Hz). They are inversely related by the equation f = 1/T. Understanding these concepts is crucial for analyzing the motion of oscillating systems, such as the ball attached to the spring.
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Spring Constant

The spring constant (k) is a measure of a spring's stiffness, defined by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from the equilibrium position (F = -kx). The spring constant is essential for determining the dynamics of oscillation, including the angular frequency, which is given by the formula ω = √(k/m), where m is the mass attached to the spring.
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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

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?

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Textbook Question

A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 m/s. Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

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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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Textbook Question

An object is undergoing SHM with period 0.900 s and amplitude 0.320 m. At t = 0 the object is at x = 0.320 m and is instantaneously at rest. Calculate the time it takes the object to go (a) from x = 0.320 m to x = 0.160 m. (b) from x = 0.160 m to x = 0.

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