 4.5.1: Refer to the equations in the definition of simple harmonic motion ...
 4.5.2: Refer to the equations in the definition of simple harmonic motion ...
 4.5.3: Refer to the equations in the definition of simple harmonic motion ...
 4.5.4: Refer to the equations in the definition of simple harmonic motion ...
 4.5.5: Refer to the equations in the definition of simple harmonic motion ...
 4.5.6: Refer to the equations in the definition of simple harmonic motion ...
 4.5.7: Spring Motion An object is attached to a coiled spring, as in Figur...
 4.5.8: Spring Motion Repeat Exercise 7, but assume that the object is pull...
 4.5.9: Voltage of an Electrical Circuit The voltage E in an electrical cir...
 4.5.10: Voltage of an Electrical Circuit For another electrical circuit, th...
 4.5.11: Particle Movement Write the equation and then determine the amplitu...
 4.5.12: Spring Motion The height attained by a weight attached to a spring ...
 4.5.13: Pendulum Motion What are the period P and frequency T of oscillatio...
 4.5.14: Pendulum Motion In Exercise 13, how long should the pendulum be to ...
 4.5.15: Spring Motion The formula for the up and down motion of a weight on...
 4.5.16: Spring Motion (See Exercise 15.) A spring with spring constant k = ...
 4.5.17: Spring Motion The position of a weight attached to a spring is s1t2...
 4.5.18: Spring Motion The position of a weight attached to a spring is s1t2...
 4.5.19: Spring Motion A weight attached to a spring is pulled down 3 in. be...
 4.5.20: Spring Motion A weight attached to a spring is pulled down 2 in. be...
 4.5.21: A weight on a spring has initial position s102 and period P. (a) To...
 4.5.22: A weight on a spring has initial position s102 and period P. (a) To...
 4.5.23: A weight on a spring has initial position s102 and period P. (a) To...
 4.5.24: A weight on a spring has initial position s102 and period P. (a) To...
 4.5.25: A note on a piano has given frequency F. Suppose the maximum displa...
 4.5.26: A note on a piano has given frequency F. Suppose the maximum displa...
 4.5.27: A note on a piano has given frequency F. Suppose the maximum displa...
 4.5.28: A note on a piano has given frequency F. Suppose the maximum displa...
 4.5.29: Consider the spring in Figure 46, but assume that because of fricti...
 4.5.30: Consider the spring in Figure 46, but assume that because of fricti...
 4.5.31: Consider the spring in Figure 46, but assume that because of fricti...
 4.5.32: Consider the spring in Figure 46, but assume that because of fricti...
 4.5.33: Consider the damped oscillatory function s1x2 = 5e0.3x cos px. (a)...
 4.5.34: Consider the damped oscillatory function s1x2 = 10ex sin 2px. (a) ...
Solutions for Chapter 4.5: Harmonic Motion
Full solutions for Trigonometry  11th Edition
ISBN: 9780134217437
Solutions for Chapter 4.5: Harmonic Motion
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: 11. Since 34 problems in chapter 4.5: Harmonic Motion have been answered, more than 23329 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780134217437. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.5: Harmonic Motion includes 34 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.