Classical Mechanics by R. Douglas Gregory

By R. Douglas Gregory

Gregory's Classical Mechanics is an important new textbook for undergraduates in arithmetic and physics. it's a thorough, self-contained and hugely readable account of a topic many scholars locate tricky. The author's transparent and systematic kind promotes a great figuring out of the topic; every one notion is prompted and illustrated by way of labored examples, whereas challenge units offer lots of perform for figuring out and strategy. computing device assisted difficulties, a few compatible for tasks, also are integrated. The publication is dependent to make studying the topic effortless; there's a common development from center subject matters to extra complex ones and tough issues are handled with specific care. A subject of the e-book is the significance of conservation ideas. those seem first in vectorial mechanics the place they're proved and utilized to challenge fixing. They reappear in analytical mechanics, the place they're proven to be concerning symmetries of the Lagrangian, culminating in Noether's theorem.

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22 Chapter 1 The algebra and calculus of vectors Problems on Chapter 1 Answers and comments are at the end of the book. Harder problems carry a star (∗). 1 . 1 In terms of the standard basis set {i, j , k}, a = 2 i − j − 2k, b = 3 i − 4 k and c = i − 5 j + 3 k. Find 3 a + 2 b − 4 c and | a − b|2 . Find | a |, | b| and a · b. Deduce the angle between a and b. Find the component of c in the direction of a and in the direction of b. Find a×b, b×c and (a×b)×(b×c). Find a · (b× c) and (a× b) · c and verify that they are equal.

16 Chapter 1 The algebra and calculus of vectors formula is a×(b×c) = (a · c) b − (a · b) c. 8) Since the vector product is anti-commutative and non-associative, it is wise to use this formula exactly as it stands. 9 Using triple products Expand the expression (a×b) · (c×d) in terms of scalar products. 6 VECTOR FUNCTIONS OF A SCALAR VARIABLE In practice, the value of a vector quantity often depends on a scalar variable such as the time t. For example, if A is the label of a particle moving through space, then its position vector a (relative to a fixed origin O) will vary with time, that is, a = a(t).

Find (i) the distance O P when t = 0, (ii) the velocity of P when t = 1, (iii) the acceleration of P when t = 2. Solution In this solution we will make use of the rules for differentiation of sums and products involving vector functions of the time. 6. (i) When t = 0, r = −3 i + 4 j so that O P = | r | = 5. (ii) Relative to the reference frame F , the unit vectors {i, j , k} are constant and so their time derivatives are zero. The velocity v of P is therefore v = d r/dt = 4t i + 4 j + (3t 2 + 4t) k.

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