/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 A block rides on a piston that i... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 19

A block rides on a piston that is moving vertically with simple harmonic motion. (a) If the SHM has period \(1.0 \mathrm{~s}\), at what amplitude of motion will the block and piston separate? (b) If the piston has an amplitude of \(5.0 \mathrm{~cm}\), what is the maximum frequency for which the block and piston will be in contact continuously?

Short Answer

Expert verified
(a) Amplitude > 24.8 cm; (b) Frequency 鈮� 2.23 Hz.

Step by step solution

01

Understand Simple Harmonic Motion (SHM)

The piston follows an SHM, characterized by the equation \( y(t) = A \, \sin(\omega t) \), where \( y \) is the displacement, \( A \) is the amplitude, and \( \omega \) is the angular frequency \( \omega = 2\pi f \). The period \( T \) is \( 1.0 \) s, so \( \omega = \frac{2\pi}{1.0} = 2\pi \text{ rad/s}.\)
02

Determine the Condition for Separation

The block will separate from the piston when the acceleration of the piston exceeds the acceleration due to gravity, i.e., when \( a_{max} > g \). For SHM, the maximum acceleration \( a_{max} = A\omega^2 \). Therefore, the separation condition is \( A(2\pi)^2 > g \).
03

Solve for Amplitude in Part (a)

Using the relationship \( A(2\pi)^2 > 9.8 \, \text{m/s}^2 \), solve for \( A \). Thus, \( A > \frac{g}{(2\pi)^2} = \frac{9.8}{4\pi^2} \approx 0.248 \, \text{m} \). Convert this to centimeters: \( A > 24.8 \, \text{cm} \).
04

Solve for Frequency in Part (b)

Given an amplitude of \( 5.0 \, \text{cm} \), set \( A\omega^2 \leq g \) to ensure continuous contact. Rearrange the equation: \( \omega^2 \leq \frac{g}{0.05} \). Calculate \( \omega \): \( \omega \leq \sqrt{\frac{9.8}{0.05}} \approx 14 \, \text{rad/s} \). Convert to frequency \( f \): \( f \leq \frac{\omega}{2\pi} \approx \frac{14}{2\pi} \approx 2.23 \, \text{Hz} \).

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Key Concepts

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

Amplitude
In simple harmonic motion (SHM), the amplitude refers to the maximum distance the object moves from its equilibrium position. Amplitude is often denoted by the symbol \( A \) and is measured in meters or centimeters.
Understanding amplitude is crucial because it's linked to the amount of energy in the motion.
  • The larger the amplitude, the more energy the system has.
  • In SHM, the amplitude determines the maximum stretch or compression.
For the block riding on a piston to not separate, we need to understand the amplitude constraints. If separated, the amplitude must be larger than a certain threshold, calculated by considering gravitational acceleration \( g \). In this context, if \( A > 24.8 \text{ cm} \), the block will separate from the piston.
Frequency
The frequency of Simple Harmonic Motion (SHM) represents how often the motion cycles are completed in one second. It's denoted by \( f \) and measured in hertz (Hz).
Frequency is closely tied to an object's angular frequency \( \omega \) because \( f = \frac{\omega}{2\pi} \).
  • High frequency means more cycles per second.
  • The lower the frequency, the slower the oscillation.
To ensure the block remains in contact with the piston, the frequency needs to be controlled. With an amplitude of 5.0 cm, the maximum frequency allowing continuous contact is approximately 2.23 Hz.
Acceleration
Acceleration in SHM describes how quickly an object's velocity changes over time. In our context, maximum acceleration \( a_{max} \) must not exceed gravity's pull for the block to remain intact with the piston:
\[ a_{max} = A \omega^2 \]
  • For separation, \( a_{max} > g \).
  • Choosing an appropriate amplitude ensures the acceleration doesn't cause separation.
By ensuring \( a_{max} \leq g \), we can prevent separation. A high or low acceleration affects whether the block stays on the piston, reflecting how forces interact during vertical SHM.
Period
Period \( T \) is the duration it takes to complete one full cycle of motion. It's the inverse of frequency: \( T = \frac{1}{f} \), given in seconds.
In the piston-block system, the period provides fundamental insight into the timing of oscillations.
  • Quick completion indicates a short period and potentially high frequency.
  • A long period suggests fewer cycles in a set time frame.
The given period of 1.0 s aligns perfectly with the need to calculate other parameters like angular frequency and ensure coherence in motion, keeping the entire system's dynamics balanced.
Separation Condition
Separation occurs when the block loses contact with the piston.
It's crucial to understand this condition to predict and manage the motion so that separation doesn't occur.
  • The key metric is ensuring the piston's maximum acceleration doesn't exceed gravitational acceleration.
  • Ensuring \( A(2\pi)^2 > g \) means exceeding that threshold causes separation.
By managing amplitude, frequency, and acceleration factors, separation can be controlled effectively, maintaining the integrity of the SHM setup in a practical setting.

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Most popular questions from this chapter

In Fig. \(15-39\), block 2 of mass \(2.0 \mathrm{~kg}\) oscillates on the end of a spring in SHM with a period of 20 \(\mathrm{ms}\). The block's position is given by \(x=(1.0 \mathrm{~cm}) \cos (\omega t+\pi / 2) .\) Block 1 of mass \(4.0 \mathrm{~kg}\) slides toward block 2 with a velocity of magnitude \(6.0\) \(\mathrm{m} / \mathrm{s}\), directed along the spring's length. The two blocks undergo a completely inelastic collision at time \(t=5.0 \mathrm{~ms}\). (The duration of the collision is much less than the period of motion.) What is the amplitude of the SHM after the collision?

What is the frequency of a simple pendulum \(2.0 \mathrm{~m}\) long (a) in a room, (b) in an elevator accelerating upward at a rate of \(2.0\) \(\mathrm{m} / \mathrm{s}^{2}\), and \((\mathrm{c})\) in free fall?

The tip of one prong of a tuning fork undergoes SHM of frequency \(1000 \mathrm{~Hz}\) and amplitude \(0.40 \mathrm{~mm}\). For this tip, what is the magnitude of the (a) maximum acceleration, (b) maximum velocity, (c) acceleration at tip displacement \(0.20 \mathrm{~mm}\), and (d) velocity at tip displacement \(0.20 \mathrm{~mm}\) ?

The amplitude of a lightly damped oscillator decreases by \(3.0 \%\) during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?

A \(5.00 \mathrm{~kg}\) object on a horizontal frictionless surface is attached to a spring with \(k=1000 \mathrm{~N} / \mathrm{m}\). The object is displaced from equilibrium \(50.0 \mathrm{~cm}\) horizontally and given an initial velocity of \(10.0 \mathrm{~m} / \mathrm{s}\) back toward the equilibrium position. What are (a) the motion's frequency, (b) the initial potential energy of the block-spring system, (c) the initial kinetic energy, and (d) the motion's amplitude?
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