Classical Dynamics of Particles and Systems
5th Edition
ISBN: 9780534408961
Author: Stephen T. Thornton, Jerry B. Marion
Publisher: Cengage Learning
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Textbook Question
Chapter 1, Problem 1.2P
Prove Equations 1.10 and 1.11 from trigonometric considerations.
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Equations 1-3 are shown below. Substitute the final expression of Eq.3 into Eq. 1 and apply the product rule of calculus to obtain three terms on the right-hand side.
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Chapter 1 Solutions
Classical Dynamics of Particles and Systems
Ch. 1 - Prob. 1.1PCh. 1 - 1.2. Prove Equations 1.10 and 1.11 from...Ch. 1 - Prob. 1.3PCh. 1 - Show
(a) (AB)t = BtAt (b) (AB)−1 = B−1 A−1
Ch. 1 - Prob. 1.5PCh. 1 - Prob. 1.6PCh. 1 - Consider a unit cube with one corner at the origin...Ch. 1 - Prob. 1.8PCh. 1 - For the two vectors
find
A − B and |A –...Ch. 1 - A particle moves in a plane elliptical orbit...
Ch. 1 - Prob. 1.11PCh. 1 - Let a, b, c be three constant vectors drawn from...Ch. 1 - X is an unknown vector satisfying the following...Ch. 1 - Prob. 1.14PCh. 1 - Prob. 1.15PCh. 1 - What surface is represented by r a = const, that...Ch. 1 - Obtain the cosine law of plane trigonometry by...Ch. 1 - Obtain the sine law of plane trigonometry by...Ch. 1 - Prob. 1.19PCh. 1 - 1-20. Show that
Ch. 1 - Show (see also Problem 1–11) that
Ch. 1 - Prob. 1.22PCh. 1 - Use the εijk notation and derive the identity
(A ×...Ch. 1 - Prob. 1.24PCh. 1 - Find the components of the acceleration vector a...Ch. 1 - Prob. 1.26PCh. 1 - If r and are both explicit functions of time,...Ch. 1 - Show that
Ch. 1 - Prob. 1.29PCh. 1 - Prob. 1.30PCh. 1 - Show that
(a)
(b)
(c)
Ch. 1 - Show that (2arr+2brr)dt=ar2+br2+const. where r is...Ch. 1 - Show that (rrrrr2)dt=rr+C where C is a constant...Ch. 1 - Prob. 1.34PCh. 1 - Prob. 1.35PCh. 1 - Prob. 1.36PCh. 1 - Prob. 1.37PCh. 1 - Prob. 1.38PCh. 1 - A plane passes through the three points (x, y, z)...Ch. 1 - For what values of a are the vectors A = 2ai − 2j...
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- 1.1 Give a physical interpretation of what is meant by the curl of a vector. 1.2 Suppose a vector function A is given by A = -y i + x j and another vector function B is given by B = x j. Calculate (i) the curl of A: and (ii) the curl of B: 1.3 In which direction are the curls pointing? Hence what can you say about their divergence and why?arrow_forwardProblem 1.45. As an illustration of why it matters which variables you hold fixec when taking partial derivatives, consider the following mathematical example. Le w = ry and z = yz. (a) Write w purely in terms of x and z, and then purely in terms of y and z. (b) Compute the partial derivatives and and show that they are not equal. (Hint: To compute (ðu/dz)y, use i formula for w in terms of r and y, not z. Similarly, compute (ðu/ar}: from a formula for w in terms of only z and z.)arrow_forwardQuestion related to Quantum Mechanics : Problem 1.15arrow_forward
- Rectangular coordinates of a point are given by (2, y) and its polar coordinates are given by (r, π/6) . Find y and r.arrow_forwardThe nearest neutron star (a collapsed star made primarily of neutrons) is about 3.00 × 1018 m away from Earth. Given that the Milky Way galaxy (Fig. P1.63) is roughly a disk of diameter ~ 1021 m and thickness ~ 1019 m, estimate the number of neutron stars in the Milky Way to the nearest order of magnitude.arrow_forwardHi, I got 1.270 as my answer. Could you please double check your answer? I did, 1/sin(52) = 1.270arrow_forward
- Problem 1.15 Calculate the divergence of the following vector functions: (a) va = x² + 3xz²ŷ - 2xz2.arrow_forward(a) Express the spherical unit vectors ê, ê, in terms of the Cartesian unit vectors ✰, ŷ, 2 (that is, derive Eq. 1.64 of Griffiths). Also work out the inverse formulas, giving ✰, ŷ, 2 in terms of f, 0, $ (and 0, $). Calculate af/00 and af/ap, and express them in terms of spherical unit vectors. (b) Express the cylindrical unit vectors ŝ, , 2 in terms of the Cartesian unit vec- tors î, ŷ, 2 (that is, derive Eq. 1.75 of Griffiths). Also work out the inverse formulas, giving x, ŷ, 2 in terms of ŝ, $, 2 (and ). Show that af/0 = $.arrow_forwardConsider the bivariate dataset for variable Minnesota, (X) and variable Rhode Island, (Y) given by the table. Rhode Group Minnesota Island А 2.24 2.98 B 2.73 2.42 C 3.80 2.97 D 3.29 2.32 E 3.90 2.25 F 3.92 3.97 a. Find the value of the slope of the regression line:arrow_forward
- 1.36 Expand log x in powers of (x – 1) by Taylor's series. -arrow_forwardLet a = (a)a + b and a-b b (b)-2ā + (c) The magnitude of a and b (d)a × b (e) Show that a. (axb) = 0 and b. (axb) = i-2j-2k and b= 61+31 + 2k; Find?arrow_forwardFind the Absolute Error (A.E) and relative error (R.E). Also, determine the number of significant digits in the approximation. (i) (ii) (iii) Hints a = 5.6543612, a' = 5.5543612 b = 48.460544, b' = 47.060543 c = 64016314, c' = 51006314 1. Use 8 digits in your calculations. 2. a', b', c' are approximations to a, b, c respectively.arrow_forward
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