/*! 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 18 SHM A \(0.50 \mathrm{~kg}\) body... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 33: Problem 18

SHM A \(0.50 \mathrm{~kg}\) body oscillates in simple harmonic motion on a spring that, when extended \(2.0 \mathrm{~mm}\) from its equilibrium, has an \(8.0 \mathrm{~N}\) restoring force. (a) What is the angular frequency of oscillation? (b) What is the period of oscillation? (c) What is the capacitance of an \(L C\) circuit with the same period if \(L\) is chosen to be \(5.0 \mathrm{H}\) ?

Short Answer

Expert verified
(a) 89.44 rad/s, (b) 0.07 s, (c) 124 碌F

Step by step solution

01

Understanding the Problem

We need to find the angular frequency of oscillation (a), the period of oscillation (b), and the capacitance of an LC circuit with the same period given its inductance (c).
02

Calculate the Spring Constant (k)

Hooke's Law states that the restoring force of a spring is proportional to its displacement: \[ F = -kx \] Given: - Restoring force, \( F = 8.0 \text{ N} \) - Displacement, \( x = 2.0 \text{ mm} = 2.0 \times 10^{-3} \text{ m} \) Thus, the spring constant is calculated as: \[ k = \frac{F}{x} = \frac{8.0 \text{ N}}{2.0 \times 10^{-3} \text{ m}} = 4.0 \times 10^{3} \text{ N/m} \]
03

Calculate the Angular Frequency (\(\omega\))

The angular frequency of a spring-mass system in SHM is given by: \[ \omega = \sqrt{\frac{k}{m}} \] Where: - \( k = 4.0 \times 10^{3} \text{ N/m} \) - Mass, \( m = 0.50 \text{ kg} \) Substitute the values: \[ \omega = \sqrt{\frac{4.0 \times 10^{3} \text{ N/m}}{0.50 \text{ kg}}} = \sqrt{8.0 \times 10^{3} \text{ s}^{-2}} \approx 89.44 \text{ rad/s} \]
04

Calculate the Period of Oscillation (T)

The period of oscillation is inversely related to the angular frequency: \[ T = \frac{2\pi}{\omega} \] From the calculated angular frequency \( \omega = 89.44 \text{ rad/s} \): \[ T = \frac{2\pi}{89.44 \text{ rad/s}} \approx 0.07 \text{ s} \]
05

Calculate the Capacitance (C)

For an LC circuit, the period \(T\) is given by: \[ T = 2\pi \sqrt{LC} \] Rearranging to find \(C\) gives: \[ C = \frac{T^2}{4\pi^2 L} \] Given: - \( T = 0.07 \text{ s} \) - Inductance, \( L = 5.0 \text{ H} \) Substitute the values: \[ C = \frac{(0.07 \text{ s})^2}{4\pi^2 (5.0 \text{ H})} \approx 1.24 \times 10^{-4} \text{ F} = 124 \mu\text{F} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

spring constant
In simple harmonic motion, the spring constant plays a vital role. This constant, often denoted as **k**, describes the stiffness of the spring. According to Hooke's Law, the force exerted by the spring (F) is directly proportional to its displacement (x) from equilibrium.
In formula terms, we write this as: F = -kx. The spring constant **k** is measured in Newtons per meter (N/m).
For our problem, we know:
  • Restoring force, F = 8.0 N
  • Displacement, x = 2.0 mm = 2.0 脳 10-3 m
To find **k**, we rearrange Hooke's Law:
k = F/x = 8.0 N / 2.0 脳 10-3 m = 4.0 脳 103 N/m. Thus, the spring constant for this spring is 4.0 脳 103 N/m.
angular frequency
The angular frequency, denoted by **蠅**, describes how quickly an object oscillates in a cycle of simple harmonic motion.
It is determined by both the spring constant (k) and the mass (m) of the oscillating object. The formula to calculate angular frequency is: 蠅 = 鈭�(k/m).
  • From our previous section, we know that k = 4.0 脳 103 N/m
  • Mass of the object, m = 0.50 kg
Using these values:
蠅 = 鈭�(4.0 脳 103 / 0.50) = 鈭�(8.0 脳 103 s-2) 鈮� 89.44 rad/s. So, the angular frequency is approximately 89.44 radians per second.
oscillation period
The period of oscillation, denoted as **T**, is the time it takes for one complete cycle of simple harmonic motion.
The period is inversely related to the angular frequency. The formula to find the period is: T = 2蟺/蠅.
  • From our calculation of angular frequency: 蠅 = 89.44 rad/s
Substituting the values:
T = 2蟺 / 89.44 rad/s 鈮� 0.07 s. Thus, the period of oscillation is approximately 0.07 seconds. This means the body takes about 0.07 seconds to complete one full oscillation.
capacitance
In an LC circuit, the capacitance (C) interacts with the inductance (L) to determine the oscillation period (T).
The period in such circuits is given by: T = 2蟺 鈭�(LC). We can rearrange this formula to solve for capacitance:
  • C = T2 / (4蟺2 L)
Given:
  • Period, T = 0.07 s
  • Inductance, L = 5.0 H
We substitute the values to get:
C = (0.07 s)2 / (4蟺2 脳 5.0 H) 鈮� 1.24 脳 10-4 F = 124 渭F. Hence, the required capacitance for the LC circuit is 124 microfarads.
Hooke's Law
Hooke's Law is fundamental for understanding simple harmonic motion involving springs.
It states that the restoring force (F) exerted by a spring is directly proportional to its displacement (x) from equilibrium. The law is expressed as: F = -kx, where k is the spring constant.
This law helps us calculate how much force is required to displace a spring by a certain distance. For instance, if we stretch or compress a spring, Hooke's Law tells us how the force produced changes with the displacement. It's essential for understanding how energy is stored in springs and how they behave in oscillatory systems such as the one discussed in our exercise.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

At What Frequency (a) At what frequency would a \(6.0 \mathrm{mH}\) inductor and a \(10 \mu \mathrm{F}\) capacitor have the same reactance? (b) What would the reactance be? (c) Show that this frequency would be the natural frequency of an oscillating circuit with the same \(L\) and \(C\).

Energy Is Dissipated In an \(R L C\) circuit such as that of assume that \(R=5.00 \Omega, L=60.0 \mathrm{mH} f^{\mathrm{dr}}=60.0 \mathrm{~Hz}\), and \(\left|\mathscr{E}^{\max }\right|=30.0 \mathrm{~V}\). For what values of the capacitor would the average rate at which energy is dissipated in the resistance be (a) a maximum and (b) a minimum? (c) What are these maximum and minimum energy dissipation rates? What are (d) the corresponding phase angles and (e) the corresponding power factors?

Sinusoidal Voltage , \(R=15.0 \Omega, C=4.70 \quad \mu \mathrm{F}\), and \(L=\) \(25.0 \mathrm{mH}\). The generator provides a sinusoidal voltage of \(75.0 \mathrm{~V}(\mathrm{rms})\) and frequency \(f=550 \mathrm{~Hz}\). (a) Calculate the rms current. (b) Find the rms voltages \(\Delta V_{a b}, \Delta V_{b c}\), \(\Delta V_{c d}, \Delta V_{b d}, \Delta V_{a d} \cdot\) (c) At what average rate is energy dissipated by each of the three circuit elements?

Length of Copper Wire A length of copper wire carries a current of 10 A, uniformly distributed through its cross section. Calculate the energy density of (a) the magnetic field and (b) the electric field at the surface of the wire. The wire diameter is \(2.5 \mathrm{~mm}\), and its resistance per unit length is \(3.3 \Omega / \mathrm{km}\).

Air Conditioner An air conditioner connected to a \(120 \mathrm{~V} \mathrm{rms}\) ac line is equivalent to a \(12.0 \Omega\) resistance and a \(1.30 \Omega\) inductive reactance in series. (a) Calculate the impedance of the air conditioner. (b) Find the average rate at which energy is supplied to the appliance.
See all solutions

Recommended explanations on Physics Textbooks

Thermodynamics

Read Explanation

Physics of Motion

Read Explanation

Fluids

Read Explanation

Force

Read Explanation

Fields in Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.