Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 1

Consider the function \(f(t)=2 t^{4}\). Evaluate the following integrals: (a) \(\int_{0}^{10} f(t) \hat{\delta}(t-3) d t\) (b) \(\int_{0}^{10} f(t) \dot{\bar{\delta}}(t-3) d t\) (c) \(\int_{0}^{10} f(t) \ddot{\bar{\delta}}(t-3) d t\) (d) \(\int_{0}^{10} \ddot{f}(t) \hat{\delta}(t-3) d t\)

Short Answer

Expert verified
Question: Evaluate the following integrals containing the Dirac Delta function and its derivatives: a) Evaluate \(\int_{0}^{10} f(t) \hat{\delta}(t-3) dt\) b) Evaluate \(\int_{0}^{10} f(t) \dot{\hat{\delta}}(t-3) dt\) c) Evaluate \(\int_{0}^{10} f(t) \ddot{\hat{\delta}}(t-3) dt\) d) Evaluate \(\int_{0}^{10} \ddot{f}(t) \hat{\delta}(t-3) dt\) Given function: \(f(t) = 2t^4\) Answer: a) The integral evaluates to 162. b) The integral evaluates to -216. c) The integral evaluates to 216. d) The integral evaluates to 216.

Step by step solution

01

a) Evaluate \(\int_{0}^{10} f(t) \hat{\delta}(t-3) dt\)

To evaluate the integral, we can use the property of Dirac Delta function, which is \(\int_{-\infty}^{\infty} f(t) \hat{\delta}(t-a) dt = f(a)\). We have the integral \(\int_{0}^{10} 2t^4 \hat{\delta}(t-3) dt\), and using the property mentioned, the result is just \(f(3)\). So, we can calculate \(f(3) = 2(3^4) = 2(81) = 162\). The integral evaluates to 162.
02

b) Evaluate \(\int_{0}^{10} f(t) \dot{\hat{\delta}}(t-3) dt\)

The integral involves the derivative of the Dirac Delta function, which has the property: \(\int_{-\infty}^{\infty} f(t) \dot{\hat{\delta}}(t-a) dt = -f'(a)\). First, we need to find the derivative of \(f(t)\). \(f'(t) = \frac{d}{dt}(2t^4) = 8t^3\). So, the integral evaluates to \(-\int_{0}^{10} 8t^3 \dot{\hat{\delta}}(t-3) dt = -f'(3)\), which is \(-8(3^3) = -216\). The integral evaluates to -216.
03

c) Evaluate \(\int_{0}^{10} f(t) \ddot{\hat{\delta}}(t-3) dt\)

The integral involves the second derivative of the Dirac Delta function, which has the property: \(\int_{-\infty}^{\infty} f(t) \ddot{\hat{\delta}}(t-a) dt = f''(a)\). To calculate this integral, first, we need to find the second derivative of \(f(t)\). \(f''(t) = \frac{d^2}{dt^2}(2t^4) = 24t^2\). So, the integral evaluates to \(\int_{0}^{10} 24t^2 \ddot{\hat{\delta}}(t-3) dt = f''(3)\), which is \(24(3^2) = 216\). The integral evaluates to 216.
04

d) Evaluate \(\int_{0}^{10} \ddot{f}(t) \hat{\delta}(t-3) dt\)

In this integral, we have the second derivative of \(f(t)\) and the Dirac Delta function. We've already calculated the second derivative as: \(\ddot{f}(t) = 24t^2\). Now, we can apply the main property of the Dirac Delta function again, and the integral evaluates to \(\ddot{f}(3)\), which is \(24(3^2) = 216\). The integral evaluates to 216.

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 timing device of Problem \(2.19\) is tapped to initiate motion. Determine the magnitude of the impulse required so that the motion of the device has an amplitude \(\Theta_{0}\).

Determine the time history of the response of a standard mass-spring-damper system when it is subjected to a triangular pulse of magnitude \(F_{0}\) and total duration \(2 \tau\), when the ramp-up and ramp-down times are of the same duration.

Determine the time history of the response of the system of Problem \(2.21\) when, over the interval \(0

Determine the response of a standard mass-spring-damper system when it is subjected to a sinusoidal transition of duration \(t^{*}\) to a constant load of magnitude \(F_{0}\).

Differential settlement of a segment of roadway causes a small drop of magnitude \(h\). Repairs are performed effecting a transition ramp of length \(L\) between the two segments of the roadway. A vehicle travels at constant speed \(v_{0}\) along a flat road when it encounters the drop. Assuming that the driver maintains the same horizontal speed throughout his motion, determine the vertical motion of the vehicle during and after it encounters the drop. The vehicle never leaves the roadway and the vertical speed due to gravity is negligible.
See all solutions

Recommended explanations on Physics Textbooks

Thermodynamics

Read Explanation

Fluids

Read Explanation

Energy Physics

Read Explanation

Waves Physics

Read Explanation

Dynamics

Read Explanation

Nuclear 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