Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 27

The period of oscillation of a spring-and-mass system is \(0.50 \mathrm{s}\) and the amplitude is \(5.0 \mathrm{cm} .\) What is the magnitude of the acceleration at the point of maximum extension of the spring?

Short Answer

Expert verified
Answer: The magnitude of the acceleration at the point of maximum extension of the spring is 25.13 m/s².

Step by step solution

01

Write down the known values

First, let's list down the given values: - Period of oscillation (T) = \(0.50s\) - Amplitude (A) = \(5.0cm\)
02

Calculate the angular frequency (ω)

We can find the angular frequency (ω) from the period using the relation: \(ω = \dfrac{2 \pi}{T}\) Now, plugging in the value of T: \(ω = \dfrac{2 \pi}{0.50s} = 4 \pi\:s^{-1}\)
03

Find the maximum displacement (X_max)

The maximum displacement of the mass occurs at the point of maximum extension, which is also equal to the amplitude A: \(X_{max} = 5.0 cm = 0.05 m\)
04

Calculate the maximum acceleration (a_max)

At the point of maximum extension, the mass experiences maximum acceleration. We can find the maximum acceleration using the general formula for the acceleration in Simple Harmonic Motion: \(a_{max} = -ω^2 X_{max}\) Now plug in the values of \(ω\) and \(X_{max}\): \(a_{max} = -(4\pi)^2 \times 0.05 = -25.13 m/s^2\) The negative sign indicates that the acceleration is opposite to the direction of the maximum displacement.
05

Report the magnitude of the acceleration

Since we only need to find the magnitude of the acceleration, we can ignore the negative sign: Magnitude of the acceleration at the point of maximum extension of the spring = \(25.13 \:m/s^2\)

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A steel piano wire \(\left(Y=2.0 \times 10^{11} \mathrm{Pa}\right)\) has a diameter of \(0.80 \mathrm{mm} .\) At one end it is wrapped around a tuning pin of diameter \(8.0 \mathrm{mm}\). The length of the wire (not including the wire wrapped around the tuning pin) is \(66 \mathrm{cm}\). Initially, the tension in the wire is \(381 \mathrm{N}\). To tune the wire, the tension must be increased to \(402 \mathrm{N}\). Through what angle must the tuning pin be turned?
The prong of a tuning fork moves back and forth when it is set into vibration. The distance the prong moves between its extreme positions is $2.24 \mathrm{mm}\(. If the frequency of the tuning fork is \)440.0 \mathrm{Hz},$ what are the maximum velocity and the maximum acceleration of the prong? Assume SHM.
Show, using dimensional analysis, that the frequency \(f\) at which a mass- spring system oscillates radical is independent of the amplitude \(A\) and proportional to Whim. IHint: Start by assuming that \(f\) does depend on \(A\) (to some power).]
A pendulum of length \(120 \mathrm{cm}\) swings with an amplitude of $2.0 \mathrm{cm} .\( Its mechanical energy is \)5.0 \mathrm{mJ} .$ What is the mechanical energy of the same pendulum when it swings with an amplitude of \(3.0 \mathrm{cm} ?\)
A small bird's wings can undergo a maximum displacement amplitude of $5.0 \mathrm{cm}$ (distance from the tip of the wing to the horizontal). If the maximum acceleration of the wings is \(12 \mathrm{m} / \mathrm{s}^{2},\) and we assume the wings are undergoing simple harmonic motion when beating. what is the oscillation frequency of the wing tips?
See all solutions

Recommended explanations on Physics Textbooks

Scientific Method Physics

Read Explanation

Force

Read Explanation

Modern Physics

Read Explanation

Kinematics Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Mechanics and Materials

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.

Sign-up for free