In case you haven't studied any differential equations

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.

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 worry about such niceties, see Problem 1.16.]

(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 .

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.

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 \(\mathrm{r}(t)=\hat{\mathrm{x}} R \cos (\omega t)+\hat{\mathrm{y}} R \sin (\omega t)\) Find the particle's velocity and acceleration. What are the magnitude and direction of the acceleration? Relate your results to well-known properties of uniform circular motion. Text Transcription: omega r(t) = hat x R cos (omega t) + hat y R sin (omega t)

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.

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

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.

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.

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 your choice of axes. Therefore you can choose axes so that r points along the x axis and s lies in the xy plane.]

(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 scalar product r r, as defined by (1.7), is the same for any choice of orthogonal axes. Use this to prove that r s, as defined by (1.7), is the same for any choice of orthogonal axes. [Hint: Consider the length of the vector r + s]

(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.

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 so- called law of sines, that a sin a sin /3 sin y Figure 1.15 Triangle for Problem 1.18.

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

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.

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).

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 8.

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.

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 to rewrite the equation as df/f = dt and then integrate both sides.] How many arbitrary constants does the general solution contain? [Your answer should illustrate the important general theorem that the solution to any nth-order differential equation (in a very large class of "reasonable" equations) contains n arbitrary constants.]

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

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 across the floor. (a) Write down the x and y coordinates of the puck as functions of time as seen from my inertial frame. (Use x and y axes pointing east and north respectively.) Now consider two more observers, the first at rest in a frame 8' that travels with constant velocity v due east relative to 8, the second at rest in a frame 8" that travels with constant acceleration due east relative to S. (All three frames coincide at the moment when I kick the puck, and 8" is at rest relative to 8 at that same moment.) (b) Find the coordinates x', y' of the puck and describe the puck's path as seen from 8'. (c) Do the same for 8". Which of the frames is inertial?

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 horizontal turntable, rotating with constant angular velocity co. I lean over and shove a frictionless puck so that it slides across the turntable, straight through the center. The puck is subject to zero net force and, as seen from my inertial frame, travels in a straight line. Describe the puck's path as observed by someone sitting at rest on the turntable. This requires careful thought, but you should be able to get a qualitative picture. For a quantitative picture, it helps to use polar coordinates; see Problem 1.46.

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.

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

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 collides with the stationary object 2. The two objects stick together and move off with common velocity v'. Use conservation of momentum to find v' in terms of v, m1, and m2.

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 contains, you can focus your attention on just two of them. (Call them 1 and 2.) The law of conservation of momentum says that if there are no external forces on this pair of particles, then their total momentum must be constant. Use this to prove that F12 = F21.]

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 B(r 1, = 47r s2 47r s2 where s = r1 r2 is the vector pointing from r2 to r1. (The first of these you should recognize as Coulomb's law.) If F712 and Fmi2ag denote the electric and magnetic forces on a charge q1 at r1 with velocity v1, show that Fr2ag < (v v2/c2)Fie21. This shows that in the non- relativistic domain it is legitimate to ignore the magnetic force between two moving charges.

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 segments of the loops, then according to the BiotSavart law, the force on dr1 due to dr2 is it. /1/2 di% x (dr2 x g) 47r s2 where s = r1 r2. The force F12 is found by integrating this around both loops. You will need to use the "B AC CAB" rule to simplify the triple product.]

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 interparticle forces are central; that is, Fap acts along the line joining particles ce and $. A full discussion of angular momentum is given in Chapter 3.]

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 the time for the golf ball to return to the ground and how far it travels in that time.

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 give the bundle's position in flight as a function of time t. (b) How far before the raft (measured horizontally) must the pilot drop the bundle if it is to hit the raft? What is this distance if vo = 50 m/s, h = 100 m, and g ti 10 m/s2? (c) Within what interval of time (+At) must the pilot drop the bundle if it is to land within +10 m of the raft?

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 starting point?

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 the z axis normal to the board. You now kick a frictionless puck that is resting at 0 so that it slides across the board with initial velocity (vox, voy, 0). Write down Newton's second law using the given coordinates and then find how long the puck takes to return to the floor level and how far it is from 0 when it does so.

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 these axes and find the ball's position as a function of time. Show that the ball lands a distance R = 2v 2 sin 0 cos(0 0)1(g cost 0) from its launch point. Show that for given vo and 0, the maximum 0 possible range up the inclined plane is R. = vo /[g(1 + sin OA

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 largest possible value of 0 if r (t) is to increase throughout the ball's flight? [Hint: Using your solution to part (a) you can write down r2 as x2 + y2, and then find the condition that r2 is always increasing.]

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.

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.

(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 the time derivatives and cb.

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.)

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 result. It explains why, in two-dimensional polars, has to be in the direction of and vice versa. It also shows that the speed of a charged particle in a magnetic field is constant, since the acceleration is perpendicular to the velocity.

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 was launched along the axis 0 = 0 at t = 0.) (b) Now write down the polar coordinates r', 0' of the puck as measured by an observer (frame S') at rest on the turntable. (Choose these coordinates so that 0 and 0' coincide at t = 0.) Describe and sketch the path seen by this second observer. Is the frame S' inertial?

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' has Cartesian coordinates (x, y, 0). Then ? and ? are defined as the two-dimensional polar coordinates of P' in the xy plane, while z is the third Cartesian coordinate, unchanged. (a) Make a sketch to illustrate the three cylindrical coordinates. Give expressions for ?, ?, z in terms of the Cartesian coordinates x, y, z. Explain in words what ? is ("? is the distance of P from "). There are many variants in notation. For instance, some people use r instead of ?. Explain why this use of r is unfortunate. (b) Describe the three unit vectors and write the expansion of the position vector r in terms of these unit vectors. (c) Differentiate your last answer twice to find the cylindrical components of the acceleration of the particle. To do this, you will need to know the time derivatives of and . You could get these from the corresponding two-dimensional results (1.42) and (1.46), or you could derive them directly as in 1.48.

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.

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 use cylindrical polar coordinates (p, 0, z) for its position (Problem 1.47), then p is fixed at p = R. while 0 and z can vary at will. Write down and solve Newton's second law for the general motion of the puck, including the effects of gravity. Describe the puck's motion.

[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 differential equation for the case that the board is released from 0 = 20 degrees, using the values R = 5 m and g = 9.8 m/s2. Make a plot of 0 against time for two or three periods. (b) On the same picture, plot the approximate solution (1.57) with the same 00= 20. Comment on your two graphs. Note: If you haven't used the numerical solver before, you will need to learn the necessary syntax. For example, in Mathematica you will need to learn the syntax for "NDSolve" and how to plot the solution that it provides. This takes a bit of time, but is something that is very well worth learning.

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

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 \(\mathrm{b} \times \mathrm{c}\). Text Transcription: b = hat x + hat y c = hat x + hat z b + c, 5b + 2c, b cdot c b times c

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 \(\mathrm{b} \times \mathrm{c}\). Text Transcription: b + c, 5b - 2c, b cdot c b times c

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^2 + y^2 + z^2

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.

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 find the angle between them as in Problem 1.4.]