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Chapter 14: Problem 2

An air-track glider attached to a spring oscillates between the \(10 \mathrm{cm}\) mark and the \(60 \mathrm{cm}\) mark on the track. The glider completes 10 oscillations in 33 s. What are the (a) period, (b) frequency, (c) angular frequency, (d) amplitude, and (e) maximum speed of the glider?

Short Answer

Expert verified
The (a) period is 3.3s, (b) frequency is 0.303 Hz, (c) angular frequency is 1.903 rad/s, (d) amplitude is 0.25m, and (e) maximum speed is 0.476 m/s.

Step by step solution

01

Calculate period

To calculate the period (T), divide the total time by the number of oscillations. So, \(T = \frac{33s}{10} = 3.3s\)
02

Calculate frequency

The frequency (f) is the reciprocal of the period. So, \(f = \frac{1}{T} = \frac{1}{3.3s} = 0.303 \, Hz\)
03

Calculate angular frequency

The angular frequency (蠅) is calculated by multiplying the frequency by 2蟺. So, \(蠅 = 2蟺f = 2蟺 * 0.303Hz = 1.903 \, rad/s\)
04

Calculate amplitude

Amplitude (A) is half the total distance travelled by the glider. The total distance is 60 cm - 10 cm = 50 cm. So, \(A = \frac{50cm}{2} = 25cm = 0.25m\)
05

Calculate maximum speed

The maximum speed (v_max) is calculated by multiplying the amplitude by the angular frequency. So, \(v_{max} = A蠅 = 0.25m * 1.903 rad/s = 0.476m/s\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Period
The period of motion in Simple Harmonic Motion (SHM) is the time it takes for one complete cycle of movement. For the air-track glider, this is the time to travel from the extreme starting point, move to the opposite extreme, and return.
To find the period, we divide the total time of motion by the number of oscillations. In this scenario, a glider completes 10 oscillations in 33 seconds. Thus,
\[ T = \frac{33 \, \mathrm{s}}{10} = 3.3 \, \mathrm{s}. \]
The period is crucial as it represents the consistent time interval for repeating cycles and is inversely related to frequency.
Frequency
Frequency in SHM refers to the number of oscillations made in a unit of time, typically measured in seconds. It describes how often the oscillation occurs.
For our glider example, frequency can be determined as the inverse of the period. Mathematically expressed as,
\[ f = \frac{1}{T} = \frac{1}{3.3 \, \mathrm{s}} = 0.303 \, \mathrm{Hz}. \]
Where "Hz" denotes hertz, the standard unit of frequency equivalent to one cycle per second.
Frequency informs us about the rapidity of oscillations, with higher frequencies indicating more cycles occurring in the same amount of time.
Angular Frequency
Angular frequency, denoted by \( \omega \), relates to how quickly the oscillator's angle changes in radians per second. It provides a bridge between frequency and circular motion concepts.
Angular frequency is calculated by multiplying frequency by \(2\pi\), as follows:
\[ \omega = 2\pi f = 2\pi \times 0.303 \, \mathrm{Hz} = 1.903 \, \mathrm{rad/s}. \]
The unit "rad/s" stands for radians per second. Angular frequency is critical in understanding the motion's dynamics, particularly in systems modeled as rotating or oscillating in a circle.
Amplitude
Amplitude in SHM signifies the maximum extent of movement from the equilibrium or central position. It measures the farthest distance the glider reaches on either side of its mean position.
To find amplitude, consider that the glider moves between the 10 cm and 60 cm marks, a total distance of 50 cm. The amplitude is half this distance:
\[ A = \frac{50 \, \mathrm{cm}}{2} = 25 \, \mathrm{cm} = 0.25 \, \mathrm{m}. \]
Amplitude is vital as it directly influences the energy in the system. Larger amplitudes mean more energy and larger motion range.
Maximum Speed
The maximum speed in SHM refers to the speed of the glider as it passes through the equilibrium position, where this speed is highest.
To find it, multiply the amplitude by the angular frequency:
\[ v_{\text{max}} = A\omega = 0.25 \, \mathrm{m} \times 1.903 \, \mathrm{rad/s} = 0.476 \, \mathrm{m/s}. \]
Maximum speed is a critical indicator of the system's kinetic energy, showcasing the glider's highest possible velocity during the motion cycle.
Understanding maximum speed helps in determining the potential for kinetic energy transfers and the overall motion dynamics of the glider.

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Most popular questions from this chapter

On your first trip to Planet X you happen to take along a \(200 \mathrm{g}\) mass, a 40 -cm-long spring, a meter stick, and a stopwatch. You're curious about the free-fall acceleration on Planet X, where ordinary tasks seem easier than on earth, but you can't find this information in your Visitor's Guide. One night you suspend the spring from the ceiling in your room and hang the mass from it. You find that the mass stretches the spring by \(31.2 \mathrm{cm} .\) You then pull the mass down \(10.0 \mathrm{cm}\) and release it. With the stopwatch you find that 10 oscillations take 14.5 s. Can you now satisfy your curiosity?

The 15 g head of a bobble-head doll oscillates in \(\mathrm{SHM}\) at a frequency of \(4.0 \mathrm{Hz}\) a. What is the spring constant of the spring on which the head is mounted? b. Suppose the head is pushed \(2.0 \mathrm{cm}\) against the spring, then released. What is the head's maximum speed as it oscillates? c. The amplitude of the head's oscillations decreases to \(0.5 \mathrm{cm}\) in \(4.0 \mathrm{s}\). What is the head's damping constant?

A block attached to a spring with unknown spring constant oscillates with a period of \(2.0 \mathrm{s}\). What is the period if a. The mass is doubled? b. The mass is halved? c. The amplitude is doubled? d. The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.

The earth's free-fall acceleration varies from \(9.78 \mathrm{m} / \mathrm{s}^{2}\) at the \(64 .\) equator to \(9.83 \mathrm{m} / \mathrm{s}^{2}\) at the poles, both because the earth is rotating and it's not a perfect sphere. A pendulum whose length is precisely \(1.000 \mathrm{m}\) can be used to measure \(g .\) Such a device is called a gravimeter a. How long do 100 oscillations take at the equator? b. How long do 100 oscillations take at the north pole? c. Is the difference between your answers to parts a and b measurable? What kind of instrument could you use to measure the difference? d. Suppose you take your gravimeter to the top of a high mountain peak near the equator. There you find that 100 oscillations take \(201.0 \mathrm{s}\). What is \(g\) on the mountain top?

A \(1.0 \mathrm{kg}\) block is attached to a spring with spring constant 16 N/m. While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of \(40 \mathrm{cm} / \mathrm{s}\) What are a. The amplitude of the subsequent oscillations? b. The block's speed at the point where \(x=\frac{1}{2} A ?\)
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