/*! 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 43 A simple pendulum is made from a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 43

A simple pendulum is made from a 0.65-m-long string and a small ball attached to its free end. The ball is pulled to one side through a small angle and then released from rest. After the ball is released, how much time elapses before it attains its greatest speed?

Short Answer

Expert verified
0.40 seconds before the pendulum reaches its greatest speed.

Step by step solution

01

Understanding the Problem

We have a simple pendulum with a length of 0.65 meters, and we need to find the time it takes for the pendulum to reach its maximum speed after being released from rest.
02

Identify the Greatest Speed Condition

The pendulum reaches its greatest speed at its lowest point, where all the potential energy has been converted to kinetic energy. For a simple pendulum, this occurs at the midpoint of its oscillation.
03

Use the Pendulum Period Formula

The period of a simple pendulum (the time for one complete oscillation back and forth) is given by the formula: \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity, which is approximately 9.81 m/s虏.
04

Calculate the Period

Substitute the given length into the formula: \( T = 2\pi \sqrt{\frac{0.65}{9.81}} \). Calculate this to find the total period of the pendulum's oscillation.
05

Calculate Time to Maximum Speed

Since the maximum speed is at the lowest point, which is one-quarter of the period from the start, the time elapsed to reach this point is \( \frac{T}{4} \). Calculate this value.
06

Final Calculation

Compute \( T = 2\pi \sqrt{\frac{0.65}{9.81}} \), then divide by 4 to find \( \frac{T}{4} \). Perform the arithmetic to find the time elapsed.

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.

Pendulum Period
The period of a simple pendulum is a crucial concept that tells us how long it takes for the pendulum to swing back and forth once. It is determined by the formula \[T = 2\pi \sqrt{\frac{L}{g}}\]where:
  • \( T \) is the period of oscillation.
  • \( L \) is the length of the pendulum.
  • \( g \) is the acceleration due to gravity, roughly 9.81 m/s虏 on Earth.
The period is directly proportional to the square root of the pendulum's length, meaning longer pendulums take more time to complete one oscillation.
Interestingly, the mass of the pendulum doesn't affect the period, assuming the angle is small.
Potential Energy
Potential energy in a pendulum is highest at the endpoints of its swing. At these points, it is momentarily at rest. This energy is stored due to the pendulum's position above its lowest point. The potential energy \[U = mgh\]where:
  • \( m \) is the mass of the pendulum.
  • \( g \) is the acceleration due to gravity.
  • \( h \) is the height from the lowest point.
As the pendulum swings downward, its potential energy decreases while its kinetic energy increases, showing the conversion of energy.
Kinetic Energy
Kinetic energy is greatest when the pendulum passes through its lowest point. Here, its speed is highest, and all potential energy has been converted into kinetic energy.
The kinetic energy of the pendulum is given by the formula:\[K = \frac{1}{2}mv^2\]where:
  • \( m \) is the mass.
  • \( v \) is the velocity of the pendulum.
The transition of energy in a pendulum between potential and kinetic is a classical example of conservation of energy.
Oscillation
An oscillation in a simple pendulum includes swinging from one side to the other and returning to the starting position. It is marked by periodic motion that repeats at regular intervals.
Each complete movement back and forth is one oscillation. The time taken for the pendulum to return to its initial point is known as the period.
The keys to understanding pendulum oscillations include:
  • The symmetry of motion: each swing has equal time durations.
  • Energy exchanges: potential and kinetic energy transition smoothly.
  • The consistent period: calculated using the pendulum period formula.
Understanding oscillation helps to comprehend how pendulums behave predictably.

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 hand exerciser utilizes a coiled spring. A force of \(89.0 \mathrm{N}\) is required to compress the spring by \(0.0191 \mathrm{m} .\) Determine the force needed to compress the spring by \(0.0508 \mathrm{m}\).

A \(0.60-\mathrm{kg}\) metal sphere oscillates at the end of a vertical spring. As the spring stretches from 0.12 to \(0.23 \mathrm{m}\) (relative to its unstrained length), the speed of the sphere decreases from 5.70 to \(4.80 \mathrm{m} / \mathrm{s}\). What is the spring constant of the spring?

A heavy-duty stapling gun uses a 0.140 -kg metal rod that rams against the staple to eject it. The rod is attached to and pushed by a stiff spring called a "ram spring" \((k=32000 \mathrm{N} / \mathrm{m})\). The mass of this spring may be ignored. The ram spring is compressed by \(3.0 \times 10^{-2} \mathrm{m}\) from its unstrained length and then released from rest. Assuming that the ram spring is oriented vertically and is still compressed by \(0.8 \times 10^{-2} \mathrm{m}\) when the downward-moving ram hits the staple, find the speed of the ram at the instant of contact.

A simple pendulum is swinging back and forth through a small angle, its motion repeating every \(1.25 \mathrm{s}\). How much longer should the pendulum be made in order to increase its period by \(0.20 \mathrm{s} ?\)

A person bounces up and down on a trampoline, while always staying in contact with it. The motion is simple harmonic motion, and it takes 1.90 s to complete one cycle. The height of each bounce above the equilibrium position is \(45.0 \mathrm{cm} .\) Determine (a) the amplitude and (b) the angular frequency of the motion. (c) What is the maximum speed attained by the person?
See all solutions

Recommended explanations on Physics Textbooks

Famous Physicists

Read Explanation

Physical Quantities And Units

Read Explanation

Waves Physics

Read Explanation

Electrostatics

Read Explanation

Electricity

Read Explanation

Electric Charge Field and Potential

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.