Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 65

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 \(3.0 \mathrm{cm} ?\)

Short Answer

Expert verified
Answer: The mechanical energy of the pendulum when its amplitude changes to 3.0 cm is 7.5 mJ.

Step by step solution

01

Understand the relationship between amplitude and mechanical energy

The mechanical energy of a simple pendulum is the sum of its kinetic and potential energy at any point in its swing. At the maximum amplitude, the pendulum has only potential energy (when it is momentarily at rest) which can be written as: \(E_{p} = mgh\) However, at the new amplitude, this potential energy increases, and the mechanical energy should also increase proportionally. Hence, we can write the relationship between the amplitude and mechanical energy as: \(E_{1} \propto h_1\) and \(E_{2} \propto h_2\) Dividing the two equations, we get: \(\frac{E_2}{E_1} = \frac{h_2}{h_1}\)
02

Determine the changes in amplitude

The initial amplitude \(h_1 = 2.0 \, \mathrm{cm}\) and the final amplitude \(h_2 = 3.0 \, \mathrm{cm}\). The change in amplitude is: \(\Delta h = h_2 - h_1 = 3.0 \, \mathrm{cm} - 2.0 \, \mathrm{cm} = 1.0 \, \mathrm{cm}\)
03

Find the ratio of the new amplitude to the old amplitude

Divide the final amplitude \(h_2\) by the initial amplitude \(h_1\): \(\frac{h_2}{h_1} = \frac{3.0 \, \mathrm{cm}}{2.0 \, \mathrm{cm}}\)
04

Calculate the new mechanical energy

We know the initial mechanical energy \(E_1 = 5.0 \, \mathrm{mJ}\), and we just found the ratio \(\frac{h_2}{h_1}\). Now, multiply \(E_1\) by the ratio \(\frac{h_2}{h_1}\) to get the new mechanical energy \(E_2\): \(E_2 = E_1 \cdot \frac{h_2}{h_1} = 5.0 \, \mathrm{mJ} \cdot \frac{3.0 \, \mathrm{cm}}{2.0 \, \mathrm{cm}}\) \(E_2 = 7.5 \, \mathrm{mJ}\) So, the mechanical energy of the pendulum when it swings with an amplitude of \(3.0 \, \mathrm{cm}\) is \(7.5 \, \mathrm{mJ}\).

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

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.
An anchor, made of cast iron of bulk modulus \(60.0 \times 10^{9} \mathrm{Pa}\) and of volume \(0.230 \mathrm{m}^{3},\) is lowered over the side of the ship to the bottom of the harbor where the pressure is greater than sea level pressure by \(1.75 \times 10^{6} \mathrm{Pa} .\) Find the change in the volume of the anchor.
The amplitude of oscillation of a pendulum decreases by a factor of 20.0 in \(120 \mathrm{s}\). By what factor has its energy decreased in that time?
A pendulum (mass \(m\), unknown length) moves according to \(x=A\) sin $\omega t .\( (a) Write the equation for \)v_{x}(t)$ and sketch one cycle of the \(v_{x}(t)\) graph. (b) What is the maximum kinetic energy?
Martin caught a fish and wanted to know how much it weighed, but he didn't have a scale. He did, however, have a stopwatch, a spring, and a \(4.90-\mathrm{N}\) weight. He attached the weight to the spring and found that the spring would oscillate 20 times in 65 s. Next he hung the fish on the spring and found that it took 220 s for the spring to oscillate 20 times. (a) Before answering part (b), determine if the fish weighs more or less than \(4.90 \mathrm{N}\). (b) What is the weight of the fish?
See all solutions

Recommended explanations on Physics Textbooks

Linear Momentum

Read Explanation

Geometrical and Physical Optics

Read Explanation

Classical Mechanics

Read Explanation

Engineering Physics

Read Explanation

Quantum Physics

Read Explanation

Fundamentals of Physics

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