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Chapter 11: Problem 51

An object on a spring vibrates in simple harmonic motion at a frequency of \(4.0 \mathrm{~Hz}\) and an amplitude of \(8.0 \mathrm{~cm}\). If the mass of the object is \(0.20 \mathrm{~kg}\), the spring constant is (a) \(40 \mathrm{~N} / \mathrm{m}\) (b) \(87 \mathrm{~N} / \mathrm{m}\) (c) \(126 \mathrm{~N} / \mathrm{m}\) (d) \(160 \mathrm{~N} / \mathrm{m}\)

Short Answer

Expert verified
(c) 126 \mathrm{~N/m}

Step by step solution

01

Understanding the Problem

We are given the frequency of the vibration as 4.0 Hz, the amplitude as 8.0 cm, and the mass of the object as 0.20 kg. We need to find the spring constant \(k\) of the spring using these values.
02

Recall the Formula for Spring-Mass System

In a simple harmonic motion system, the angular frequency \(\omega\) is related to the spring constant \(k\) and the mass \(m\) by the formula \(\omega = \sqrt{\frac{k}{m}}\). We can use this relationship to determine \(k\).
03

Relate Angular Frequency to Given Frequency

The angular frequency \(\omega\) is related to the frequency \(f\) by the formula \(\omega = 2\pi f\). Taking the given frequency \(f = 4.0\, \mathrm{Hz}\), calculate \(\omega\):\[\omega = 2\pi \times 4.0\, \mathrm{Hz} = 8\pi\, \mathrm{rad/s}.\]
04

Substitute Values into the Spring-Mass Formula

Substitute the calculated \(\omega\), given mass \(m = 0.20\, \mathrm{kg}\), and solve for \(k\):\[8\pi = \sqrt{\frac{k}{0.20}}\]Squaring both sides gives:\[(8\pi)^2 = \frac{k}{0.20}.\]
05

Solve for the Spring Constant

First, calculate \((8\pi)^2\):\[(8\pi)^2 = 64\pi^2 = 64 \times 9.87 \approx 631.1.\]Now solve for \(k\):\[k = 0.20 \times 631.1 \approx 126 \mathrm{~N/m}.\]
06

Choose the Correct Answer

Comparing the calculated spring constant of approximately \(126 \mathrm{~N/m}\) to the given options, the answer is (c) \(126 \mathrm{~N/m}\).

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Key Concepts

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

Spring Constant Calculation
The spring constant, often represented as \(k\), is a measure of the stiffness of a spring. In simple harmonic motion, the spring constant can be determined using the properties of the system such as mass, frequency, and angular frequency.

Here's how you can calculate it:
  • First, identify the known values like mass \(m\) and frequency \(f\).
  • Use the formula for angular frequency \(\omega = 2\pi f\) to find \(\omega\).
  • Next, use the relation \(\omega = \sqrt{\frac{k}{m}}\) to solve for \(k\).
In essence, the spring constant \(k\) defines the force needed to stretch or compress the spring by a unit length. The larger the value of \(k\), the stiffer the spring.
Angular Frequency
Angular frequency, denoted as \(\omega\), is a key concept in analyzing oscillatory systems such as a mass-spring system. It describes how fast something is rotating or oscillating.

It is calculated using the formula:
  • \(\omega = 2\pi f\), where \(f\) is the frequency.
In simple harmonic motion, angular frequency links together the linear oscillations to rotational concepts because one full oscillation corresponds to a rotation of \(2\pi\) radians. This makes it useful for harmonizing periodic motion with rotational motion. In the context of simple harmonic motion, it plays a crucial role in determining how the amplitude and period of oscillation relate to the force constants involved.
Mass-Spring System
A mass-spring system is a classic example of simple harmonic motion. It consists of a mass attached to a spring, which can oscillate back and forth when displaced from its equilibrium position.

Key characteristics of a mass-spring system:
  • The mass \(m\): determines the inertia and consequently the dynamic behavior of the system.
  • The spring constant \(k\): describes the spring's stiffness and directly affects the system's oscillation characteristics.
  • Upon stretching or compressing the spring, the system experiences a restoring force proportional to the displacement and the spring constant \(k\).
The balance between the mass's inertia and the spring's stiffness dictates how quickly the system returns to equilibrium. It also influences the oscillations' period and energy exchange between kinetic and potential forms.
Frequency and Amplitude
Frequency and amplitude are fundamental properties describing oscillatory motion in systems such as a mass-spring setup.

**Frequency \(f\):**
  • It represents how many oscillations occur in a unit time frame, typically measured in Hertz (Hz).
  • Determined by the system's physical properties like mass and spring constant.
**Amplitude:**
  • Represents the maximum displacement from the equilibrium position.
  • Reflects how far the oscillating object moves from rest position during each cycle.
Together, frequency and amplitude determine the nature of the oscillatory motion. While frequency influences how fast the motion repeats, amplitude indicates its reach or intensity. Both are essential for predicting and analyzing wave behaviors and responses in various physical contexts.

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Most popular questions from this chapter

Which of the following expressions does not represent SHM? (a) \(A \cos \omega t\) (b) \(A \sin 2 \omega t\) (c) \(A \sin \omega t+B \cos \omega t\) (d) \(A \sin ^{2} \omega t\)

The equation of SHM is given as \(x=3 \sin 20 \pi t+4 \cos 20 \pi t\) where \(x\) is in \(\mathrm{cms}\) and \(t\) is in seconds. The amplitude is (a) \(7 \mathrm{~cm}\) (b) \(4 \mathrm{~cm}\) (c) \(5 \mathrm{~cm}\) (d) \(3 \mathrm{~cm}\)

Which of the following properties of a wave does not change with a change in medium? (a) Frequency (b) Wavelength (c) Velocity (d) Amplitude

A string of length \(L\) is stretched by \(L / 20\) and the speed of transverse waves along it is \(v\). The speed of wave when it is stretched by \(L / 10\) will be (assume that Hooke's law is applicable) (a) \(2 v\) (b) \(v / \sqrt{2}\) (c) \(\sqrt{2} v\) (d) \(4 v\)

The amplitude of a wave disturbance propagating in the positive \(Y\) -direction is given by \(y=\frac{1}{1+x^{2}}\) at \(t=0\) and \(y=\frac{1}{\left[1+(x-1)^{2}\right]}\) at \(t=2 \mathrm{~s}\) where, \(x\) and \(y\) are in \(\mathrm{m}\). If the shape of the wave disturbance does not change during the propagation, what is the velocity of the wave? (a) \(1 \mathrm{~m} / \mathrm{s}\) (b) \(1.5 \mathrm{~m} / \mathrm{s}\) (c) \(0.5 \mathrm{~m} / \mathrm{s}\) (d) \(2 \mathrm{~m} / \mathrm{s}\)
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