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Chapter 7: Problem 15

This problem deals with a mass \(m\) on a spring (with constant \(k\) ) that receives an impulse \(p_{0}=m v_{0}\) at time \(t=0\). Show that the initial value problems $$ m x^{\prime \prime}+k x=0 ; \quad x(0)=0, \quad x^{\prime}(0)=v_{0} $$ and $$ m x^{\prime \prime}+k x=p_{0} \delta(t) ; \quad x(0)=0, \quad x^{\prime}(0)=0 $$ have the same solution. Thus the effect of \(p_{0} \delta(t)\) is, indeed, to impart to the particle an initial momentum \(p_{0}\).

Short Answer

Expert verified
The impulse \( p_0 \delta(t) \) imparts initial momentum, leading to identical solutions for both equations.

Step by step solution

01

Solve the Homogeneous Equation

Consider the first initial value problem (IVP):\[ m x^{\prime \prime} + k x = 0; \quad x(0) = 0, \quad x^{\prime}(0) = v_0 \]This is a standard second-order linear homogeneous differential equation. The general solution of this equation is given by:\[ x(t) = A \cos(\omega t) + B \sin(\omega t) \]where \( \omega = \sqrt{\frac{k}{m}} \).
02

Apply Initial Conditions for the First IVP

Using the initial conditions:1. \( x(0) = 0 \) implies \[ A \cos(0) + B \sin(0) = 0 \Rightarrow A = 0 \]2. \( x^{\prime}(0) = v_0 \) implies \[ \frac{d}{dt}[A \cos(\omega t) + B \sin(\omega t)]_{t=0} = v_0 \Rightarrow B \omega = v_0 \Rightarrow B = \frac{v_0}{\omega} \]Thus, the solution becomes:\[ x(t) = \frac{v_0}{\omega} \sin(\omega t) \]
03

Solve the Inhomogeneous Equation using Impulse Function

Consider the second IVP:\[ m x^{\prime \prime} + k x = p_0 \delta(t); \quad x(0) = 0, \quad x^{\prime}(0) = 0 \]This inhomogeneous equation includes the Dirac delta function. The impulse \( p_0 \delta(t) \) changes the initial momentum of the system but not the functional form of the solution after \( t = 0 \).
04

Address the Impulse Effect and Solve After t=0

The Dirac delta function \( \delta(t) \) imparts the momentum instantly at \( t = 0 \). This changes the initial condition for velocity from \( x^{\prime}(0) = 0 \) to \( x^{\prime}(0) = \frac{p_0}{m} = v_0 \).Therefore, for \( t > 0 \), the second IVP solves to the same form as in Step 1:\[ x(t) = \frac{v_0}{\omega} \sin(\omega t) \]
05

Conclusion of Solutions

Both initial value problems reduce to the equation:\[ x(t) = \frac{v_0}{\omega} \sin(\omega t) \]Thus, both problems have the same solution for \( t > 0 \), confirming that \( p_0 \delta(t) \) imparts the initial momentum \( p_0 \) to the system.

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Key Concepts

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

Mass-Spring System
In physics, a mass-spring system is a simple model used to understand motion. It consists of a weight, or mass, attached to a spring. You can think of a mass-spring system like a weight hanging from a bouncy string that stretches and compresses as it moves.

Let's break it down:
  • Mass: The mass ( m ) represents the object attached to the spring. It tries to resist motion, meaning it doesn't want to move when it doesn't have to.

  • Spring Constant: The spring constant ( k ) measures the stiffness of the spring. A high value means a really tight spring, while a low value is more like a loose, stretchy spring.

  • Oscillation: When you extend or compress the spring and let go, the mass will move back and forth. This kind of motion is called oscillation. It is repetitive and regular.

For a system obeying Hooke's Law, the force on the mass is kx , where x is the displacement. Newton's second law leads to the differential equation m x'' + kx = 0, describing simple harmonic motion. Understanding this helps in predicting the system's behavior when forces act upon it.
Dirac Delta Function
The Dirac delta function \(\delta(t)\) is not a typical function you can plot like a line or curve. Instead, it behaves in a special, abstract way. Imagine it as a spike at a single point along a timeline, representing an "instant effect."

  • Impulse Effect: The delta function applies an instantaneous impulse to a system. Think of it as applying a precisely timed hammer tap that occurs instantaneously at t=0 .

  • Mathematical Representation: In equations, \(\int_{-\infty}^{\infty} f(t) \delta(t) \, dt = f(0)\) shows how the delta function acts like a "sifter," pulling out the value of another function at t=0 .

  • Usefulness in Systems: In a mass-spring system, p_0 \(\delta(t)\) models an impulse that instantly changes the momentum. It doesn't physically move the mass at t=0 but gives it a jolt that starts it in motion.

Understanding the Dirac delta function is crucial because it helps model real-world events that happen abruptly and significantly affect system behavior, often as initial jolts that set systems in motion.
Initial Value Problems
Initial value problems (IVPs) are a staple in differential equations. They focus on finding a solution to an equation when we know the state of the system at a starting point, typically time zero.

Consider an IVP as a story: You start with beginning conditions and seek how the narrative unfolds. Here's how it works:
  • Initial Conditions: Define the starting point of the system, such as position x(0) = 0 and velocity x'(0) = v_0 for the mass-spring system. These values define the system's starting "state."
  • Differential Equation: Governs how the system changes over time. For example, m x'' + kx = 0 guides the mass's motion in a spring.

  • Finding the Solution: The goal is to determine the function describing behavior, which usually involves calculus. You want a function like x(t) that fits the starting conditions and equation.

IVPs are all about maps to the future. They take a known start and show the journey to other times, drawing paths linking math to real-world situations like waves, electricity, or in our case, the movement of springs.

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Most popular questions from this chapter

Find the Laplace transforms of the functions given in Problems. $$ f(t)=0 \text { if } t<1 ; f \overline{(t)} \equiv t \text { if } t \geq 1 $$

Use Laplace transforms to solve the initial value problems in Problem. $$ x^{(4)}-x=0 ; x(0)=1, x^{\prime}(0)=x^{\prime \prime}(0)=x^{(3)}(0)=0 $$

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Find the Laplace transforms of the functions given in Problems. $$ f(t)=\cos \frac{1}{2} \pi t \text { if } 3 \leqq t \leqq 5 ; f(t)=0 \text { if } t<3 \text { or if } t>5 $$

Use Laplace transforms to solve the initial value problems. $$ x^{\prime \prime}+4 x=\cos t: x(0)=0=x^{\prime}(0) $$
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