Schaum series free download pdf

More than , copies sold in its first 2 editions; Over 93, students enrolled; Translated into 12 languages; Corresponds to standard college economics courses; Use with most macroeconomics texts; Includes a new chapter on economic growth.

This updated edition reflects changes and developments in the field of macroeconomics. This is a solved-problems outline for standard undergraduate and graduate economics courses in colleges and MBA programs. New topics included are national income accounting, lags in demand stabilization, the Phillips curve and monetarist macroeconomics.

Over four-hundred solved and supplementary problems are included. Best of all, you'll receive free website support for additional guidance and need-to-know updates. The unique Cross-Platform format adds outstanding value: students can study the whole program in print, online, or on a mobile device. Practice with MCAT-style questions on every topic Textbook-quality illustrations to enhance understanding Focuses tightly on topics tested on the MCAT About the Cross-Platform format: The Cross-Platform format provides a fully-comprehensive print, online, and mobile program: Entire instructional content available in print and digital form Personalized study plan and daily goals Powerful analytics to assess test readiness Flashcards, games, and social media for additional support About the Authors George Hademenos, Ph.

Candice McCloskey Campbell, Ph. Shaun Murphree, Ph. Jennifer M. Warner, Ph. Amy B. Wachholz, Ph. Kathy A. Zahler, MS, is a widely published test-prep au. With the review books, you get textbook-quality scientific diagrams, concise summaries of all the important concepts, and abundant practice questions. Madrid Mexico City Milan Schaum Spiegel, Murray R. Schaum's outline of theory and problems of Fourier analysis.

Schaum's outline Scha New York Chicago San Francisco. Lisbon London Madrid Mexico City. Thus, Substituting these values into 1 , we obtain The general solution is therefore, This is a second-order equation for x t with It follows from Eq.

The general solution is Solve In? We first write the differential equation in standard form, with unity as the coefficient of the highest derivative. The one exception is when ihc general solution is ihe homogeneous solution; lhai is, when the dilTerenlial equation under eon si derail on is iise.

The jicneral solution of the differential equation is given in Problem 1 I. I as Therefore. Solve The general solution of the differential equation is given in Problem Solve From Problem The solution to the initial-value problem is The system is in its equilibrium position when it is at rest. The mass is set in motion by one or more of the following means: displacing the mass from its equilibrium position, providing it with an initial velocity, or subjecting it to an external force F i.

A steel ball weighing Ib is suspended from a spring, whereupon the spring is stretched 2 ft from its natural length. The applied force responsible for the 2-ft displacement is the weight of the ball, Ib. For convenience, we choose the downward direction as the positive direction and take the origin to be the center of gravity of the mass in the equilibrium position. We assume that the mass of the spring is negligible and can be neglected and that air resistance, when present, is proportional to the velocity of the mass.

Note that the restoring force Fs always acts in a direction that will tend to return the system to the equilibrium position: if the mass is below the equilibrium position, then x is positive and -kx is negative; whereas if the mass is above the equilibrium position, then x is negative and -kx is positive. The force of gravity does not explicitly appear in We automatically compensated for this force by measuring distance from the equilibrium position of the spring. If one wishes to exhibit gravity explicitly, then distance must be measured from the bottom end of the natural length of the spring.

The current 7 flowing through the circuit is measured in amperes and the charge q on the capacitor is measured in coulombs. Kirchhojfs loop law: The algebraic sum of the voltage drops in a simple closed electric circuit is zero. It is known that the voltage drops across a resistor, a capacitor, and an inductor are respectively RI, HC q, and L dlldt where q is the charge on the capacitor. The voltage drop across an emf is —E t. Thus, from Kirchhoff s loop law, we have The relationship between q and 7 is Substituting these values into The second initial condition is obtained from Eq.

Thus, An expression for the current can be gotten either by solving Eq. See Problems Such a body experiences two forces, a downward force due to gravity and a counter force governed by: Archimedes' principle: A body in liquid experiences a buoyant upward force equal to the weight of the liquid displaced by that body.

Equilibrium occurs when the buoyant force of the displaced liquid equals the force of gravity on the body. Figure depicts the situation for a cylinder of radius r and height 77 where h units of cylinder height are submerged at equilibrium. At equilibrium, the volume of water displaced by the cylinder is 7tr2h, which provides a buoyant force of 7tr2hp that must equal the weight of the cylinder mg.

If the cylinder is raised out of the water by x t units, as shown in Fig. The downward or negative force on such a body remains mg but the buoyant or positive force is reduced to Jtr2[h - x t ]p. It now follows from Newton's second law that Substituting For buoyancy problems defined by Eq. For electrical circuit problems, the independent variable x is replaced either by q in Eq.

For damped motion, there are three separate cases to consider, depending on whether the roots of the associated characteristic equation see Chapter 9 are 1 real and distinct, 2 equal, or 3 complex conjugate. These cases are respectively classified as 1 overdamped, 2 critically damped, and 3 oscillatory damped or, in electrical problems, underdamped.

A steady-state motion or current is one that is not transient and does not become unbounded. Free undamped motion defined by Eq. Here c1, c2, and ft are constants with ft often referred to as circular frequency. The natural frequency j'is and it represents the number of complete oscillations per time unit undertaken by the solution. The period of the system of the time required to complete one oscillation is Equation The ball is started in motion with no initial velocity by displacing it 6 in above the equi- librium position.

The motion is free and undamped. Equation Find an expression for the motion of the mass, assuming no air resistance. The equation of motion is governed by Eq. Differentiating 2 , we obtain whereupon, and 2 simplifies to as the position of the mass at any time t.

Determine the circular frequency, natural frequency, and period for the simple harmonic motion described in Problem Circular frequency: Natural frequency: Period: A kg mass is attached to a spring, stretching it 0. The mass is started in motion from the equilibrium position with an initial velocity of 1 ml sec in the upward direction. Find the subsequent motion, if the force due to air resistance is i N.

Find the subsequent motion of the mass if the force due to air resistance is -2ilb. Find the subsequent motion of the mass, if the force due to air resistance is -lilb. Show that the types of motions that result from free damped problems are completely determined by the quantity a2 — 4 km.

The corresponding motions are, respectively, overdamped, critically damped, and oscillatory damped. Since the real parts of both roots are always negative, the resulting motion in all three cases is transient.

Find the subsequent motion of the mass if the force due to air resistance is iN. The equation of motion, These terms are the transient part of the solution. Assuming no air resistance, find the subsequent motion of the weight. This phenomenon is called pure resonance. It is due to the forcing function F t having the same circular frequency as that of the associated free undamped system.

Write the steady-state motion found in Problem The steady-state displacement is given by 2 of Problem Substituting the given quantities into Eq. Hence, As in Problem Solve Problem Substituting the values given in Problem Therefore, and as before. Note that although the current is completely transient, the charge on the capacitor is the sum of both transient and steady-state terms.

An RCL circuit connected in series has a resistance of 5 ohms, an inductance of 0. Find an expression for the current flowing through this circuit if the initial current and the initial charge on the capacitor are both zero. Substituting this value into 2 and simplifying, we obtain as before Determine the circular frequency, the natural frequency, and the period of the steady-state current found in Problem The current is given by 3 of Problem Write the steady-state current found in Problem Determine whether a cylinder of radius 4 in, height 10 in, and weight 15 Ib can float in a deep pool of water of weight density Let h denote the length in feet of the submerged portion of the cylinder at equilibrium.

Determine an expression for the motion of the cylinder described in Problem In the context of Fig. Determine whether a cylinder of diameter 10 cm, height 15 cm, and weight Let h denote the length in centimeters of the submerged portion of the cylinder at equilibrium. Let h denote the length of the submerged portion of the cylinder at equilibrium. A solid cylinder partially submerged in water having weight density Determine the diameter of the cylinder if it weighs 2 Ib.

We are given 0. The prism is set in motion by displacing it from its equilibrium position see Fig. Determine the differential equation governing the subsequent motion of this prism. For the prism depicted in Fig. By Archimedes' principle, this buoyant force at equilibrium must equal the weight of the prism mg; hence, We arbitrarily take the upward direction to be the positive x-direction.

If the prism is raised out of the water by x t units, as shown in Fig. It now follows from Newton's second law that Substituting 1 into this last equation, we simplify it to Fig. A lb weight is suspended from a spring and stretches it 2 in from its natural length. Find the spring constant. A mass of 0. It is then set into motion by stretching the spring 2 in from its equilibrium position and releasing the mass from rest.

Find the position of the weight at any time t if there is no external force and no air resistance. Find the position of the mass at any time t if there is no external force and no air resistance.

A lb weight is attached to a spring, stretching it 8 ft from its natural length. Find the subsequent motion of the weight, if the medium offers negligible resistance. Determine a the circular frequency, b the natural frequency, and c the period for the vibrations described in Problem Find the solution to Eq.

A -slug mass is hung onto a spring, whereupon the spring is stretched 6 in from its natural length. Find the subsequent motion of the mass, if the force due to air resistance is —2x Ib. A -j-slug mass is attached to a spring so that the spring is stretched 2 ft from its natural length. The mass is started in motion with no initial velocity by displacing it yft in the upward direction. Find the subsequent motion of the mass, if the medium offers a resistance of —4x Ib.

The mass is set into motion by displacing it 6 in below its equilibrium position with no initial velocity. Find the subsequent motion of the mass, if the force due to the medium is —4x Ib.

Find the subsequent motion of the mass if the surrounding medium offers a resistance of -4iN. Find the subsequent motion of the mass, if the force due to air resistance is —4x Ib. A lb weight is attached to a spring whereupon the spring is stretched 1. Find the subsequent motion of the weight if the surrounding medium offers a negligible resistance.

A lb weight is attached to a spring whereupon the spring is stretched 2 ft and allowed to come to rest. Find the subsequent motion of the weight if the surrounding medium offers a resistance of —2x Ib. Write the steady-state portion of the motion found in Problem Find the subsequent motion of the mass if the surrounding medium offers a resistance of —3x N. Assuming no initial current and no initial charge on the capacitor, find expressions for the current flowing through the circuit and the charge on the capacitor at any time t.

Assuming no initial current and no initial charge on the capacitor, find an expression for the current flowing through the circuit at any time t. Determine the steady-state current in the circuit described in Problem An RCL circuit connected in series with a resistance of 16 ohms, a capacitor of 0.

Assuming no initial current and no initial charge on the capacitor, find an expression for the charge on the capacitor at any time t. Determine the steady-state charge on the capacitor in the circuit described in Problem Find the subsequent steady-state current in the circuit. Initial conditions are not needed. Find the steady-state current in the circuit.

Hint Initial conditions are not needed. Determine the equilibrium position of a cylinder of radius 3 in, height 20 in, and weight 57rlb that is floating with its axis vertical in a deep pool of water of weight density Find an expression for the motion of the cylinder described in Problem Write the harmonic motion of the cylinder described in Problem Determine the equilibrium position of a cylinder of radius 2 ft, height 4 ft, and weight Ib that is floating with its axis vertical in a deep pool of water of weight density Determine the equilibrium position of a cylinder of radius 30 cm, height cm, and weight 2.

Find the general solution to Eq. Hint: Use the results of Problem The box is set into motion by displacing it x0 units from its equilibrium position and giving it an initial velocity of v0. Determine the differential equation governing the subsequent motion of the box. Determine a the period of oscillations for the motion described in Problem II all the elements are numbers.

Ihen the matrix is called a constant matrix. Matrices will prove to be very helpful in several ways. For example, we can recast higher-order differential equations into a sjslem of first-order differential equations using matrices see Chapter In particular, the first matrix is a constant matrix, whereas the last two are not.

A matrix is square if it has the same number of rows and columns. The third matrix given in Example I5. I is a vector. That is, Matrix addition is both associative and commutalue. Matrix multiplication is associative and distributes over addition; in general, however, it is not commutative. Theorem Cayley—Hamilton theorem. Any square matrix satisfies its own characteristic equation. That is, if then Solved Problems Find 3A - B for the matrices given in Problem Find 2A - B 2 for the matrices given in Problem But Therefore, the cancellation law is not valid for matrix multiplication.

Find Ax if Find Find J A dt for A as given in Problem Find the eigenvalues of A? Verify the Cayley-Hamilton theorem for the matrix of Problem Find a AB and b BA. Find A2. Find A7. FindB 2. Find a CD and b DC. Find a Ax and b xA. Find AC. Find the characteristic equation and eigenvalues of A. Find the characteristic equation and eigenvalues of B. Find the characteristic equation and eigenvalues of 3A. Find the characteristic equation and the eigenvalues of C. Determine the multiplicity of each eigenvalue.

Find the characteristic equation and the eigenvalues of D. Find for A as given in Problem Find Adt for A as given in Problem The infinite series However, it follows with some effort from Theorem I.

Thus: Theorem Furthermore, if X; is an eigeinalue of multiplicity k. When com- puting the various derivatives in Method of computation: For each eigenvalue A,, of A? When this is done for each eigenvalue, the set of all equations so obtained can be solved for a0, «i, These values are then substituted into Eq. From Eq. Substituting these values successively into Eq.

Substituting these values successively into It follows from Theorem From Eqs. It now follows from Theorem Now, according to Eq. Consider the following second- order differential equation: We see that! The method of reduction is as follows.

Step 1. Rewrite Thus, where and Step 2. From the mathematical foundations to fluid mechanics and viscoelasticity, this guide covers all the fundamentals—plus it shows you how theory is applied. This is the study guide to choose if you want to ace continuum mechanics! This ideal review for the thousands of students who enroll in thermodynamics courses Thermodynamics for Engineers is intended to help engineering students in their understanding of the discipline in a more concise, ordered way than that used in standard textbooks, which are often filled with extraneous material never addressed in the classroom.

This edition conforms to the more user-friendly, pragmatic approach now used in most classes. The outline provides practice sets to allow students to work through the theory they've learned. Material is organized by discrete topics such as gas cycles, vapor cycles, and refrigeration cycles.

Practice tests simulate the quizzes and tests given in class. There are also fully solved problems, as well as questions of the type that appear on the engineers' qualifying exam.

This new edition boasts problem-solving videos available online and embedded in the ebook version. Fortunately for you, there's Schaum's Outlines.