Classical Mechanics 0th Edition solutions

Classical Mechanics 0
Given the two vectors \(\mathrm{b}=\hat{\mathrm{x}}+\hat{\mathrm{y}}\) and \(\mathrm{c}=\hat{\mathrm{x}}+\hat{\mathrm{z}}\) find \(\mathrm{b}+\mathrm{c},\ 5\mathrm{b}+2\mathrm{c},\mathrm{\ b}\cdot\mathrm{c}\), and \(\mathrm{b} \times \mathrm{c}\).

Classical Mechanics 0
Two vectors are given as b = (1, 2, 3) and c = (3, 2, 1). (Remember that these statements are just a compact way of giving you the components of the vectors.) Find \(\mathrm{b}+\mathrm{c},\ 5\mathrm{b}-2\mathrm{c},\mathrm{\ b}\cdot\mathrm{c}\), and \(\mathrm{b} \times \mathrm{c}\).

Classical Mechanics 0
By applying Pythagoras's theorem (the usual two-dimensional version) twice over, prove that the length r of a three-dimensional vector r = (x, y, z) satisfies \(r^{2}=x^{2}+y^{2}+z^{2}\). Text Transcription: r^2 = x

Classical Mechanics 0
One of the many uses of the scalar product is to find the angle between two given vectors. Find the angle between the vectors b = (1, 2, 4) and c = (4, 2, 1) by evaluating their scalar product.

Classical Mechanics 0
Find the angle between a body diagonal of a cube and any one of its face diagonals. [Hint: Choose a cube with side 1 and with one corner at O and the opposite corner at the point (1, 1, 1). Write down the vector that represents a body diagonal and another that represents a face diagonal, and then fin

Classical Mechanics 0
By evaluating their dot product, find the values of the scalar s for which the two vectors b = x + s'Sr and c = x sS7 are orthogonal. (Remember that two vectors are orthogonal if and only if their dot product is zero.) Explain your answers with a sketch.

Classical Mechanics 0
Prove that the two definitions of the scalar product r s as rs cos 9 (1.6) and E ris, (1.7) are equal. One way to do this is to choose your x axis along the direction of r. [Strictly speaking you should first make sure that the definition (1.7) is independent of the choice of axes. If you like to wor

Classical Mechanics 0
(a) Use the definition (1.7) to prove that the scalar product is distributive, that is, r (u + v) = r u + r v. (b) If r and s are vectors that depend on time, prove that the product rule for differentiating products applies to r s, that is, that ds dr dt (r s) r dt + dt s .

Classical Mechanics 0
In elementary trigonometry, you probably learned the law of cosines for a triangle of sides a, b, and c, that c2 = a2 b2 2ab cos 9, where 9 is the angle between the sides a and b. Show that the law of cosines is an immediate consequence of the identity (a + b)2 = a2 b2 + 2a b.

Classical Mechanics 0
A particle moves in a circle (center O and radius R) with constant angular velocity \(\omega\) counter-clockwise. The circle lies in the xy plane and the particle is on the x axis at time t = 0. Show that the particle's position is given by \(\mathrm{r}(t)=\hat{\mathrm{x}} R

Classical Mechanics 0
The position of a moving particle is given as a function of time t to be r(t) = xb cos(wt) + STc sin(wt), where b, c, and w are constants. Describe the particle's orbit.

Classical Mechanics 0
The position of a moving particle is given as a function of time t to be r(t) = xb cos(wt) + S7c sin(wt) + ivot

Classical Mechanics 0
Let u be an arbitrary fixed unit vector and show that any vector b satisfies b2 (u b)2 + (u x b)2. Explain this result in words, with the help of a picture.

Classical Mechanics 0
Prove that for any two vectors a and b, la + bi < (a + b). [Hint: Work out la 1312 and compare it with (a + b)2.] Explain why this is called the triangle inequality.

Classical Mechanics 0
Show that the definition (1.9) of the cross product is equivalent to the elementary definition that r x s is perpendicular to both r and s, with magnitude rs sin 0 and direction given by the right-hand rule. [Hint: It is a fact (though quite hard to prove) that the definition (1.9) is independent of

Classical Mechanics 0
(a) Defining the scalar product r s by Equation (1.7), r s = E risi , show that Pythagoras's theorem implies that the magnitude of any vector r is r = r. (b) It is clear that the length of a vector does not depend on our choice of coordinate axes. Thus the result of part (a) guarantees that the scala

Classical Mechanics 0
(a) Prove that the vector product r x s as defined by (1.9) is distributive; that is, that r x (u v) = --- (r x u) (r x v). (b) Prove the product rule d ds dr dt (rxs)=rx dt at xs. Be careful with the order of the factors.

Classical Mechanics 0
The three vectors a, b, c are the three sides of the triangle ABC with angles a, ,8, y as shown in Figure 1.15. (a) Prove that the area of the triangle is given by any one of these three expressions: area = -21a x bl = -2 lb X CI = -2 x al. (b) Use the equality of these three expressions to prove the

Classical Mechanics 0
If r, v, a denote the position, velocity, and acceleration of a particle, prove that d[a (v x r)] = a (v x r). dt

Classical Mechanics 0
The three vectors A, B, C point from the origin 0 to the three corners of a triangle. Use the result of Problem 1.18 to show that the area of the triangle is given by (area of triangle) = 2I(B x C) + (C x A) + (A x B)1.

Classical Mechanics 0
A parallelepiped (a six-faced solid with opposite faces parallel) has one corner at the origin 0 and the three edges that emanate from 0 defined by vectors a, b, c. Show that the volume of the parallelepiped is a (b x c).

Classical Mechanics 0
The two vectors a and b lie in the xy plane and make angles a and with the x axis. (a) By evaluating a b in two ways [namely using (1.6) and (1.7)] prove the well-known trig identity cos(a 0) = cos a cos 0 + sin a sin ,8. (b) By similarly evaluating a x b prove that sin(a 0) = sin a cos /3 cos a sin

Classical Mechanics 0
The unknown vector v satisfies b v = A and b x v = c, where , b, and c are fixed and known. Find v in terms of , b, and c.

Classical Mechanics 0
In case you haven't studied any differential equations before, I shall be introducing the necessary ideas as needed. Here is a simple excercise to get you started: Find the general solution of the first- order equation df/dt = f for an unknown function f(t). [There are several ways to do this. One is

Classical Mechanics 0
Answer the same questions as in Problem 1.24, but for the differential equation df/dt = 3f .

Classical Mechanics 0
The hallmark of an inertial reference frame is that any object which is subject to zero net force will travel in a straight line at constant speed. To illustrate this, consider the following: I am standing on a level floor at the origin of an inertial frame S and kick a frictionless puck due north ac

Classical Mechanics 0
The hallmark of an inertial reference frame is that any object which is subject to zero net force will travel in a straight line at constant speed. To illustrate this, consider the following experiment: I am standing on the ground (which we shall take to be an inertial frame) beside a perfectly flat

Classical Mechanics 0
Go over the steps from Equation (1.25) to (1.29) in the proof of conservation of momentum, but treat the case that N = 3 and write out all the summations explicitly to be sure you understand the various manipulations.

Classical Mechanics 0
Do the same tasks as in Problem 1.28 but for the case of four particles (N = 4).

Classical Mechanics 0
Conservation laws, such as conservation of momentum, often give a surprising amount of information about the possible outcome of an experiment. Here is perhaps the simplest example: Two objects of masses m1 and m2 are subject to no external forces. Object 1 is traveling with velocity v when it collid

Classical Mechanics 0
In Section 1.5 we proved that Newton's third law implies the conservation of momentum. Prove the converse, that if the law of conservation of momentum applies to every possible group of particles, then the interparticle forces must obey the third law. [Hint: However many particles your system contain

Classical Mechanics 0
If you have some experience in electromagnetism, you could do the following problem concerning the curious situation illustrated in Figure 1.8. The electric and magnetic fields at a point r1 due to a charge q2 at r2 moving with constant velocity v2 (with v2 << c) arel5 1 q2 \ [to q2 v 2 E(ri) = s and

Classical Mechanics 0
If you have some experience in electromagnetism and with vector calculus, prove that the magnetic forces, F12 and F21, between two steady current loops obey Newton's third law. [Hints: Let the two currents be II and /2 and let typical points on the two loops be r1 and r2. If dr1 and dr2 are short seg

Classical Mechanics 0
Prove that in the absence of external forces, the total angular momentum (defined as L = ED, ra x pa) of an N-particle system is conserved. [Hints: You need to mimic the argument from (1.25) to (1.29). In this case you need more than Newton's third law: In addition you need to assume that the interpa

Classical Mechanics 0
A golf ball is hit from ground level with speed vo in a direction that is due east and at an angle 6 above the horizontal. Neglecting air resistance, use Newton's second law (1.35) to find the position as a function of time, using coordinates with x measured east, y north, and z vertically up. Find t

Classical Mechanics 0
A plane, which is flying horizontally at a constant speed vo and at a height h above the sea, must drop a bundle of supplies to a castaway on a small raft. (a) Write down Newton's second law for the bundle as it falls from the plane, assuming you can neglect air resistance. Solve your equations to gi

Classical Mechanics 0
A student kicks a frictionless puck with initial speed vo, so that it slides straight up a plane that is inclined at an angle 0 above the horizontal. (a) Write down Newton's second law for the puck and solve to give its position as a function of time. (b) How long will the puck take to return to its

Classical Mechanics 0
You lay a rectangular board on the horizontal floor and then tilt the board about one edge until it slopes at angle 0 with the horizontal. Choose your origin at one of the two corners that touch the floor, the x axis pointing along the bottom edge of the board, the y axis pointing up the slope, and t

Classical Mechanics 0
A ball is thrown with initial speed vo up an inclined plane. The plane is inclined at an angle above the horizontal, and the ball's initial velocity is at an angle 0 above the plane. Choose axes with x measured up the slope, y normal to the slope, and z across it. Write down Newton's second law using

Classical Mechanics 0
A cannon shoots a ball at an angle 6 above the horizontal ground. (a) Neglecting air resistance, use Newton's second law to find the ball's position as a function of time. (Use axes with x measured horizontally and y vertically.) (b) Let r(t) denote the ball's distance from the cannon. What is the la

Classical Mechanics 0
An astronaut in gravity-free space is twirling a mass m on the end of a string of length R in a circle, with constant angular velocity co. Write down Newton's second law (1.48) in polar coordinates and find the tension in the string.

Classical Mechanics 0
Prove that the transformations from rectangular to polar coordinates and vice versa are given by the four equations (1.37). Explain why the equation for 0 is not quite complete and give a complete version.

Classical Mechanics 0
(a) Prove that the unit vector r of two-dimensional polar coordinates is equal to = X cos 0 + 5r sin 0 (1.59) and find a corresponding expression for 4. (b) Assuming that 0 depends on the time t, differentiate your answers in part (a) to give an alternative proof of the results (1.42) and (1.46) for

Classical Mechanics 0
Verify by direct substitution that the function 0(t) = A sin(wt) + B cos(wt) of (1.56) is a solution of the second-order differential equation (1.55), if, = co20. (Since this solution involves two arbitrary constants the coefficients of the sine and cosine functions it is in fact the general solution

Classical Mechanics 0
Prove that if v(t) is any vector that depends on time (for example the velocity of a moving particle) but which has constant magnitude, then is orthogonal to v(t). Prove the converse that if is orthogonal to v (t), then is constant. [Hint: Consider the derivative of .] This is a very handy

Classical Mechanics 0
Consider the experiment of Problem 1.27, in which a frictionless puck is slid straight across a rotating turntable through the center 0. (a) Write down the polar coordinates r, 0 of the puck as functions of time, as measured in the inertial frame S of an observer on the ground. (Assume that the puck

Classical Mechanics 0
Let the position of a point P in three dimensions be given by the vector r = (x, y, z) in rectangular (or Cartesian) coordinates. The same position can be specified by cylindrical polar coordinates, ?, ?, z, which are defined as follows: Let P' denote the projection of P onto the xy plane; that is, P

Classical Mechanics 0
Find expressions for the unit vectors j), 4, and i of cylindrical polar coordinates (Problem 1.47) in terms of the Cartesian X, y, i. Differentiate these expressions with respect to time to find 01dt, clikldt, and dildt.

Classical Mechanics 0
Imagine two concentric cylinders, centered on the vertical z axis, with radii R E, where E is very small. A small frictionless puck of thickness 2E is inserted between the two cylinders, so that it can be considered a point mass that can move freely at a fixed distance from the vertical axis. If we u

Classical Mechanics 0
[Computer] The differential equation (1.51) for the skateboard of Example 1.2 cannot be solved in terms of elementary functions, but is easily solved numerically. (a) If you have access to software, such as Mathematica, Maple, or Matlab, that can solve differential equations numerically, solve the di

Classical Mechanics 0
[Computer] Repeat all of Problem 1.50 but using the initial value 00 = r/2.