/*! 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 80 A block is in SHM on the end of ... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 80

A block is in SHM on the end of a spring, with position given by \(x=x_{m} \cos (\omega t+\phi)\). If \(\phi=\pi / 5 \mathrm{rad}\), then at \(t=0\) what percentage of the total mechanical energy is potential energy?

Short Answer

Expert verified
About 65.45% of the total mechanical energy is potential energy at \(t=0\).

Step by step solution

01

Identify Given Values and Formula

The position of the block in simple harmonic motion (SHM) is given by \(x = x_m \cos(\omega t + \phi)\). We know \(\phi = \pi / 5\) radians. At \(t=0\), the position \(x\) becomes \(x = x_m \cos(\phi)\). This is the initial position of the block.
02

Calculate Initial Position

Substitute \(\phi = \pi / 5\) into the expression for \(x\):\[ x = x_m \cos\left(\frac{\pi}{5}\right) \]Calculate \(\cos(\pi/5)\) to find the initial position of the block, where \(x_m\) is the amplitude.
03

Express Potential Energy Formula

The potential energy in SHM is given by the expression:\[ U = \frac{1}{2} k x^2 \]Where \(k\) is the spring constant and \(x\) is the position at time \(t=0\).
04

Express Total Mechanical Energy Formula

The total mechanical energy \(E\) in SHM, which is constant, is given by:\[ E = \frac{1}{2} k x_m^2 \]This expression represents the maximum energy, when all energies are potential.
05

Calculate Percentage of Potential Energy

The percentage of potential energy in the total energy is:\[ \text{Percentage of Potential Energy} = \left(\frac{U}{E}\right) \times 100 \]Substitute \(x = x_m \cos(\pi/5)\), \(U = \frac{1}{2} k (x_m \cos(\pi/5))^2\), and \(E = \frac{1}{2} k x_m^2\) into the formula:\[ \text{Percentage} = \frac{(\cos(\pi/5))^2}{1} \times 100 = (\cos(\pi/5))^2 \times 100 \]Calculate \((\cos(\pi/5))^2\) to determine the final percentage.

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Key Concepts

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

Potential Energy in Simple Harmonic Motion (SHM)
Potential energy plays a crucial role in simple harmonic motion (SHM). It is the stored energy of an object when it is in a particular position within its oscillatory cycle. In the context of a block-spring system, the potential energy (U) at any given point is determined by the block's position and the spring constant. The formula used to calculate potential energy is:\[ U = \frac{1}{2} k x^2 \]Here, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position. This means that potential energy is highest when the displacement is at its maximum value, which occurs at the amplitude of the motion.
  • When the block is at its equilibrium position, its potential energy is zero because \(x = 0\).
  • As the block moves from the equilibrium position to its maximum displacement, potential energy increases.
Understanding potential energy is vital, especially when determining the distribution of energy in SHM, as different energies transform from one form to another, keeping the total energy conserved.
Total Mechanical Energy in SHM
The total mechanical energy in simple harmonic motion is a combination of both potential and kinetic energy. In an ideal system without any energy loss, this total mechanical energy (E) stays constant throughout the motion. The formula to express total mechanical energy is:\[ E = \frac{1}{2} k x_m^2 \]where \(x_m\) represents the maximum displacement, also known as the amplitude. Total mechanical energy can be visualized as being completely potential energy when the block is momentarily at rest at the maximum displacement, and entirely kinetic when it passes through the equilibrium position with maximum speed.
  • At maximum displacement: All energy is potential, \(U = E\).
  • At equilibrium position: All energy is kinetic.
Thus, understanding how total mechanical energy is shared between kinetic and potential energy helps to illustrate the energy dynamics during SHM.
Spring Constant and Its Role
The spring constant (k) is a fundamental parameter in the study of simple harmonic motion. It measures the stiffness of the spring and determines how resistant the spring is to being compressed or stretched. The higher the spring constant, the stiffer the spring. In formulas concerning SHM, such as the potential energy and total mechanical energy equations, the spring constant appears as:
  • Potential energy: \( U = \frac{1}{2} k x^2 \)
  • Total mechanical energy: \( E = \frac{1}{2} k x_m^2 \)
The spring constant directly affects the amount of energy stored as potential energy and the total mechanical energy of the system.
  • A larger spring constant results in greater potential energy for a given displacement.
  • It also impacts the frequency of oscillation, linked through the angular frequency \(\omega = \sqrt{\frac{k}{m}}\), where \(m\) is mass.
Understanding the spring constant is essential for analyzing the behavior of oscillating systems and predicting their motion in various scenarios.

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Most popular questions from this chapter

A pendulum is formed by pivoting a long thin rod about a point on the rod. In a series of experiments, the period is measured as a function of the distance \(x\) between the pivot point and the rod's center. (a) If the rod's length is \(L=2.20 \mathrm{~m}\) and its mass is \(m=22.1 \mathrm{~g}\), what is the minimum period? (b) If \(x\) is cho- sen to minimize the period and then \(L\) is increased, does the period increase, decrease, or remain the same? (c) If, instead, \(m\) is increased without \(L\) increasing, does the period increase, decrease, or remain the same?

When the displacement in SHM is one-half the amplitude \(x_{m}\), what fraction of the total energy is (a) kinetic energy and (b) potential energy? (c) At what displacement, in terms of the amplitude, is the energy of the system half kinetic energy and half potential energy?

An oscillator consists of a block of mass \(0.500 \mathrm{~kg}\) connected to a spring. When set into oscillation with amplitude \(35.0 \mathrm{~cm}\), the oscillator repeats its motion every \(0.500 \mathrm{~s}\). Find the (a) period, (b) frequency, (c) angular frequency, (d) spring constant, (e) maximum speed, and (f) magnitude of the maximum force on the block from the spring.

A \(3.0 \mathrm{~kg}\) particle is in simple harmonic motion in one dimension and moves according to the equation $$ x=(5.0 \mathrm{~m}) \cos [(\pi / 3 \mathrm{rad} / \mathrm{s}) t-\pi / 4 \mathrm{rad}] $$ with \(t\) in seconds. (a) At what value of \(x\) is the potential energy of the particle equal to half the total energy? (b) How long does the particle take to move to this position \(x\) from the equilibrium position?

\(=9\) The position function \(x=\) \((6.0 \mathrm{~m}) \cos [(3 \pi \mathrm{rad} / \mathrm{s}) t+\pi / 3 \mathrm{rad}]\) gives the simple harmonic motion of a body. At \(t=2.0 \mathrm{~s}\), what are the (a) displacement, (b) velocity, (c) acceleration, and (d) phase of the motion? Also, what are the (e) frequency and (f) period of the motion?
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