/*! 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 35 In a spring-block oscillator, th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 35

In a spring-block oscillator, the maximum speed of the block is (A) proportional to amplitude (B) proportional to the square of amplitude (C) proportional to the square root of amplitude (D) inversely proportional to the square root of amplitude

Short Answer

Expert verified
The maximum speed of the block in a spring-block oscillator is proportional to the amplitude, so the correct answer is (A) proportional to amplitude.

Step by step solution

01

Identify Relevant Equation

The question involves Maximum speed and Amplitude. We need to find the relationship between them. The velocity equation for harmonic motion in a spring-block oscillator is given by \(v = \omega\sqrt{A^2 - x^2}\), where \(v\) is velocity, \(\omega\) is angular frequency, \(A\) is amplitude, and \(x\) is displacement.
02

Determine the Maximum Speed

Consider when the speed is at a maximum, the displacement is zero \(x = 0\). Thus the equation becomes: \(v_{max} = \omega\sqrt{A^2}\). Simplifying, we get \(v_{max} = \omega A\)
03

Analyze the Result

Looking at the resulting equation, we see that the maximum speed \(v_{max}\) is proportional to the amplitude \(A\). It is not related to the square or the square root of the amplitude, and there is no inverse relationship.

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.

Harmonic Motion
Harmonic motion, which is also known as simple harmonic motion (SHM), describes the natural oscillation of an object when it's disturbed from an equilibrium position.

When a system exhibits harmonic motion, it moves back and forth in a regular, repeating pattern. This is characterized by:
  • The restoring force, which is directly proportional to the displacement from the equilibrium position.
  • An oscillation pattern that resembles a sine or cosine wave.
  • Stable amplitude and frequency.
A common example is the motion of a mass on a spring, where the system's energy switches between potential energy in the spring and kinetic energy of the moving mass. The motion is perfectly periodic, meaning it repeats at regular intervals, making it predictable and easy to model mathematically.
Spring-Block Oscillator
A spring-block oscillator is a typical setup that demonstrates harmonic motion. It consists of a block attached to a spring that can move horizontally on a frictionless surface.

This system is crucial for understanding fundamental physics concepts like energy conservation and oscillations.
  • When the block is displaced from its equilibrium by stretching or compressing the spring, it experiences a restoring force that attempts to bring it back to its original position.
  • This restoring force, according to Hooke's Law, is proportional to the displacement, which leads to harmonic motion.
  • The spring-block model is crucial because it serves as a simple representation of more complex physical systems.
The ease with which this model displays harmonic motion makes it an excellent tool for learning about oscillatory systems and the principles of mechanics.
Maximum Speed
In the context of a spring-block oscillator, the maximum speed (\(v_{max}\)) of the block is achieved when it passes through the equilibrium position.

At this point, all the potential energy stored in the spring is converted into kinetic energy. Here's how you can determine this speed:
  • Use the formula for velocity in harmonic motion: \(v = \omega\sqrt{A^2 - x^2}\), where \(\omega\) is angular frequency, \(A\) is amplitude, and \(x\) is displacement.
  • When the speed is maximum, displacement \(x\) is zero, so the equation simplifies to \(v_{max} = \omega A\).
  • This formula reveals that the maximum speed is simply the product of angular frequency and amplitude.
These points highlight how speed dynamics change during oscillation, illustrating the energy transformation within the system.
Amplitude Relationship
Amplitude in harmonic motion refers to the maximum extent of an oscillation or displacement from the equilibrium position. It plays a critical role in determining the dynamics of a spring-block oscillator.

The relationship between amplitude and maximum speed is straightforward:
  • The greater the amplitude, the higher the maximum speed, as derived from the equation \(v_{max} = \omega A\).
  • This indicates a direct proportional relationship: doubling the amplitude will double the maximum speed, assuming constant angular frequency.
  • This relationship is influential in systems design, where specific speed requirements dictate the necessary amplitude characteristics.
Understanding this relationship helps in anticipating how changes to amplitude will influence the kinetic performance of oscillatory systems.

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 toy car and a toy truck collide. If the toy truck鈥檚 mass is double the toy car鈥檚 mass, then, compared to the acceleration of the truck, the acceleration of the car during the collision will be (A) double the magnitude and in the same direction (B) double the magnitude and in the opposite direction (C) half the magnitude and in the same direction (D) half the magnitude and in the opposite direction

A person standing on a horizontal floor is acted upon by two forces: the downward pull of gravity and the upward normal force of the floor. These two forces (A) have equal magnitudes and form an action-reaction pair (B) have equal magnitudes and do not form an action-reaction pair (C) have unequal magnitudes and form an action-reaction pair (D) have unequal magnitudes and do not form an action-reaction pair

Given that the Earth's mass is \(m,\) its tangential speed as it revolves around the Sun is \(v\) , and the distance from the Sun to the Earth is \(r,\) which of the following correctly describes the work done by the centripetal force, \(W_{c}\) , in one year's time? (A) \(\mathrm{W}_{\mathrm{c}}>2 r\left(m v^{2} / r\right)\) (B) \(\mathrm{W}_{\mathrm{c}}=2 r\left(m v^{2} / r\right)\) (C) \(\mathrm{W}_{\mathrm{c}}<2 r\left(\mathrm{mv}^{2} / r\right)\) (D) Cannot be determined

Directions: Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part. (Figure is not available to copy) An experiment is conducted in which Block A with a mass of \(m_{A}\) is slid to the right across a friction less table. Block A collides with Block B, which is initially at rest, of an unknown mass and sticks to it. (a) Describe an experimental procedure that determines the velocities of the blocks before and after a collision. Include all the additional equipment you need. You may include a labeled diagram of your setup to help in your description. Indicate what measurements you would take and how you would take them. Include enough detail so that the experiment could be repeated with the procedure you provide. (b) If Block A has a mass of 0.5 kg and starts off with a speed of 1.5 m/s and the experiment is repeated, the velocity of the blocks after the collide are recorded to be 0.25 m/s. What is the mass of Block B? (c) How much kinetic energy was lost in this collision from part (b)?

N resistors \((N > 2)\) are connected in parallel with a battery of voltage \(V_{o} .\) If one of the resistors is removed from the circuit, which of the following quantities will decrease? Select two answers. (A) The voltage across any of the remaining resistors (B) The current output by the battery (C) The total power dissipated in the circuit (D) The voltage supplied by the battery
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Electricity

Read Explanation

Wave Optics

Read Explanation

Fundamentals of Physics

Read Explanation

Turning Points in Physics

Read Explanation

Electrostatics

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.