- 3.5.Problem 1: A point
- 3.5.a: Define linear velocity and give its formula.
- 3.5.1: If a point travels along a circle with constant velocity, then the ...
- 3.5.Problem 2: A point P on a circle rotates through radians in 4 seconds. Give th...
- 3.5.b: Define angular velocity and give its formula
- 3.5.2: For a point moving with uniform circular motion, the distance trave...
- 3.5.Problem 3: A bicycle
- 3.5.c: What is the relationship between linear and angular velocity?
- 3.5.3: Linear velocity is _____________ to angular velocity. A point rotat...
- 3.5.Problem 4: Repeat Example 4,
- 3.5.d: What is the relationship between linear and angular velocity?
- 3.5.4: The formula relating linear
- 3.5.Problem 5: A phonograph
- 3.5.5: s 3 ft and t 2 min 6
- 3.5.Problem 6: Repeat Example 6, but assume the diameter is 200 feet and one compl...
- 3.5.6: s 10 ft and t 2 min
- 3.5.7: s 12 cm and t 4 sec 8.
- 3.5.8: s 12 cm and t 2 sec
- 3.5.9: s 30 mi and t 2 hr 1
- 3.5.10: s 100 mi and t 4 hr
- 3.5.11: v 20 ft/sec and t 4 sec
- 3.5.12: v 10 ft/sec and t 4 sec 1
- 3.5.13: v 45 mi/hr and t 1 2 hr
- 3.5.14: v 55 mi/hr and t 1 2 hr
- 3.5.15: v 21 mi/hr and t 20 min
- 3.5.16: v 63 mi/hr and t 10 sec
- 3.5.17: 24, t 6 min 18
- 3.5.18: 12, t 3 min 1
- 3.5.19: 8, t 3sec 20.
- 3.5.20: 12, t 5sec 21.
- 3.5.21: 45, t 1.2 hr 22.
- 3.5.22: 24, t 1.8 hr 23
- 3.5.23: Rotating Light Figure 3 shows a lighthouse that is 100 feet from a ...
- 3.5.24: Rotating Light Using the diagram in Figure 3, find an equation that...
- 3.5.25: 4 rad/sec, r 2 inches, t 5 sec 2
- 3.5.26: 2 rad/sec, r 4 inches, t 5 sec 2
- 3.5.27: 3/2 rad/sec, r 4 m, t 30 sec 28
- 3.5.28: 4/3 rad/sec, r 8 m, t 20 sec 29
- 3.5.29: 10 rad/sec, r 6 ft, t 2 min 3
- 3.5.30: 15 rad/sec, r 5 ft, t 1 min F
- 3.5.31: 10 rpm
- 3.5.32: 20 rpm
- 3.5.33: 33 1 3 rpm
- 3.5.34: 16 2 3 rpm
- 3.5.35: 5.8 rpm
- 3.5.36: 7.2 rpm
- 3.5.37: Find v if r 2 inches and 5 rad/sec.
- 3.5.38: Find v if r 8 inches and 4 rad/sec.
- 3.5.39: Find if r 6 cm and v 3 cm/sec.
- 3.5.40: Find if r 3 cm and v 8 cm/sec.
- 3.5.41: Find v if r 4 ft and the point rotates at 10 rpm.
- 3.5.42: Find v if r 1 ft and the point rotates at 20 rpm.
- 3.5.43: Velocity at the Equator The earth rotates through one complete revo...
- 3.5.44: Velocity at the Equator Assuming the radius of the earth is 4,000 m...
- 3.5.45: Velocity of a Mixer Blade A mixing blade on a food processor extend...
- 3.5.46: Velocity of a Lawnmower Blade A gasoline-driven lawnmower has a bla...
- 3.5.47: Cable Cars The San Francisco cable cars travel by clamping onto a s...
- 3.5.48: Cable Cars The Los Angeles Cable Railway was driven by a 13-foot-di...
- 3.5.49: Cable Cars The Cleveland City Cable Railway had a 14-foot-diameter ...
- 3.5.50: Cable Cars The old Sutter Street cable car line in San Francisco (F...
- 3.5.51: Ski Lift A ski lift operates by driving a wire rope, from which cha...
- 3.5.52: Ski Lift A
- 3.5.53: Velocity of a Ferris Wheel Figure 7 is a model of the Ferris wheel ...
- 3.5.54: Velocity of a Ferris Wheel Use Figure 7 as a model of the Ferris wh...
- 3.5.55: Ferris Wheel For the Ferris wheel described in 53, find the height ...
- 3.5.56: Ferris Wheel For the Ferris wheel described in 54, find the height ...
- 3.5.57: Velocity of a Bike Wheel A woman rides a bicycle for 1 hour and tra...
- 3.5.58: Velocity of a Bike Wheel Find the number of revolutions per minute ...
- 3.5.59: The first gear in a single-stage gear train has 42 teeth and an ang...
- 3.5.60: The second gear in a single-stage gear train has 6 teeth and an ang...
- 3.5.61: A gear train consists of three gears meshed together (Figure 9). Th...
- 3.5.62: A two-stage gear train consists of four gears meshed together (Figu...
- 3.5.63: When Lance Armstrong blazed up Mount Ventoux in the 2002 Tour, he w...
- 3.5.64: On level ground, Lance would use a larger chainring and a smaller s...
- 3.5.65: If Lance was using his 210-millimeter-diameter chainring and pedali...
- 3.5.66: If Lance was using his 150-millimeter-diameter chainring and pedali...
- 3.5.67: Suppose Lance was using a 150-millimeter-diameter chainring and an ...
- 3.5.68: Suppose Lance was using a 210-millimeter-diameter chainring and a 4...
- 3.5.69: Magnitude of a Vector Find the magnitudes of the horizontal and ver...
- 3.5.70: Magnitude of a Vector The magnitude of the horizontal component of ...
- 3.5.71: Distance and Bearing A ship sails for 85.5 miles on a bearing of S ...
- 3.5.72: Distance and Bearing A plane flying with a constant speed of 285.5 ...
- 3.5.73: Find the linear velocity of a point moving with uniform circular mo...
- 3.5.74: Find the angular velocity of a point moving with uniform circular m...
- 3.5.75: Find the linear velocity of a point rotating at 30 revolutions per ...
- 3.5.76: A pulley is driven by a belt moving at a speed of 15.7 feet per sec...
Solutions for Chapter 3.5: Velocities
Full solutions for Trigonometry | 7th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
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!
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
= Xl (column 1) + ... + xn(column n) = combination of columns.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
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.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.