Chapter 1

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

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

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

One of the many uses of the scalar product is to find the angle between two given vectors. Find the angle between the vectors \(\mathbf{b}=(1,2,4)\) and \(\mathbf{c}=(4,2,1)\) 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.]

By evaluating their dot product, find the values of the scalar \(s\) for which the two vectors \(\mathbf{b}=\hat{\mathbf{x}}+s \hat{\mathbf{y}}\) and \(\mathbf{c}=\hat{\mathbf{x}}-s \hat{\mathbf{y}}\) 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 \(\mathbf{r} \cdot\) s as \(r s \cos \theta(1.6)\) and \(\sum r_{i} s_{i}(1.7)\) are equal. One way to do this is to choose your \(x\) axis along the direction of \(\mathbf{r}\). [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.]

In elementary trigonometry, you probably learned the law of cosines for a triangle of sides \(a, b,\) and \(c,\) that \(c^{2}=a^{2}+b^{2}-2 a b \cos \theta,\) where \(\theta\) is the angle between the sides \(a\) and \(b\). Show that the law of cosines is an immediate consequence of the identity \((\mathbf{a}+\mathbf{b})^{2}=a^{2}+b^{2}+2 \mathbf{a} \cdot \mathbf{b}\)

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

The position of a moving particle is given as a function of time \(t\) to be $$\mathbf{r}(t)=\hat{\mathbf{x}} b \cos (\omega t)+\hat{\mathbf{y}} c \sin (\omega t)$$ where \(b, c,\) and \(\omega\) are constants. Describe the particle's orbit.

The position of a moving particle is given as a function of time \(t\) to be $$\mathbf{r}(t)=\hat{\mathbf{x}} b \cos (\omega t)+\hat{\mathbf{y}} c \sin (\omega t)+\hat{\mathbf{z}} v_{\mathrm{o}} t$$ where \(b, c, v_{\mathrm{o}}\) and \(\omega\) are constants. Describe the particle's orbit.

Let \(\mathbf{u}\) be an arbitrary fixed unit vector and show that any vector \(\mathbf{b}\) satisfies $$b^{2}=(\mathbf{u} \cdot \mathbf{b})^{2}+(\mathbf{u} \times \mathbf{b})^{2}$$ Explain this result in words, with the help of a picture.

A parallelepiped (a six-faced solid with opposite faces parallel) has one corner at the origin \(O\) and the three edges that emanate from \(O\) defined by vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) Show that the volume of the parallelepiped is \(|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})|\)

The unknown vector \(\mathbf{v}\) satisfies \(\mathbf{b} \cdot \mathbf{v}=\lambda\) and \(\mathbf{b} \times \mathbf{v}=\mathbf{c},\) where \(\lambda, \mathbf{b},\) and \(\mathbf{c}\) are fixed and known. Find \(\mathbf{v}\) in terms of \(\lambda, \mathbf{b},\) and \(\mathbf{c}\)

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 firstorder equation \(d f / d t=f\) for an unknown function \(f(t) .\) [There are several ways to do this. One is to rewrite the equation as \(d f / f=d t\) 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 \(n\) th-order differential equation (in a very large class of "reasonable" equations) contains \(n\) arbitrary constants.]

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 \(\mathcal{S}\) 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 \(\mathcal{S}^{\prime}\) that travels with constant velocity \(v\) due east relative to \(\mathcal{S},\) the second at rest in a frame \(\mathcal{S}^{\prime \prime}\) that travels with constant acceleration due east relative to \(\mathcal{S}\). (All three frames coincide at the moment when I kick the puck, and \(\mathcal{S}^{\prime \prime}\) is at rest relative to \(\mathcal{S}\) at that same moment.) (b) Find the coordinates \(x^{\prime}, y^{\prime}\) of the puck and describe the puck's path as seen from \(\mathcal{S}^{\prime} .\) (c) Do the same for \(\mathcal{S}^{\prime \prime}\) Which of the frames is inertial?

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 \(m_{1}\) and \(m_{2}\) are subject to no external forces. Object 1 is traveling with velocity \(\mathbf{v}\) when it collides with the stationary object \(2 .\) The two objects stick together and move off with common velocity \(\mathbf{v}^{\prime}\). Use conservation of momentum to find \(\mathbf{v}^{\prime}\) in terms of \(\mathbf{v}, m_{1},\) and \(m_{2}\)

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 \(\mathbf{F}_{12}=-\mathbf{F}_{21}\).]

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 \(\mathbf{r}_{1}\) due to a charge \(q_{2}\) at \(\mathbf{r}_{2}\) moving with constant velocity \(\mathbf{v}_{2}\) (with \(v_{2} \ll c\) ) are \(^{15}\) $$\mathbf{E}\left(\mathbf{r}_{1}\right)=\frac{1}{4 \pi \epsilon_{\mathrm{o}}} \frac{q_{2}}{s^{2}} \hat{\mathbf{s}} \quad \text { and } \quad \mathbf{B}\left(\mathbf{r}_{1}\right)=\frac{\mu_{0}}{4 \pi} \frac{q_{2}}{s^{2}} \mathbf{v}_{2} \times \hat{\mathbf{s}}$$ where \(\mathbf{s}=\mathbf{r}_{1}-\mathbf{r}_{2}\) is the vector pointing from \(\mathbf{r}_{2}\) to \(\mathbf{r}_{1}\). (The first of these you should recognize as Coulomb's law.) If \(\mathbf{F}_{12}^{\mathrm{el}}\) and \(\mathbf{F}_{12}^{\text {mag }}\) denote the electric and magnetic forces on a charge \(q_{1}\) at \(\mathbf{r}_{1}\) with velocity \(\mathbf{v}_{1},\) show that \(F_{12}^{\operatorname{mag}} \leq\left(v_{1} v_{2} / c^{2}\right) F_{12}^{\mathrm{el}} .\) 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, \(\mathbf{F}_{12}\) and \(\mathbf{F}_{21}\), between two steady current loops obey Newton's third law. [Hints: Let the two currents be \(I_{1}\) and \(I_{2}\) and let typical points on the two loops be \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\). If \(d \mathbf{r}_{1}\) and \(d \mathbf{r}_{2}\) are short segments of the loops, then according to the Biot- Savart law, the force on \(d \mathbf{r}_{1}\) due to \(d \mathbf{r}_{2}\) is $$\frac{\mu_{0}}{4 \pi} \frac{I_{1} I_{2}}{s^{2}} d \mathbf{r}_{1} \times\left(d \mathbf{r}_{2} \times \hat{\mathbf{s}}\right)$$ where \(\mathbf{s}=\mathbf{r}_{1}-\mathbf{r}_{2} .\) The force \(\mathbf{F}_{12}\) is found by integrating this around both loops. You will need to use the " \(B A C-C A B\) " rule to simplify the triple product.]

A plane, which is flying horizontally at a constant speed \(v_{\mathrm{o}}\) 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 \(v_{\mathrm{o}}=50 \mathrm{m} / \mathrm{s}\) \(h=100 \mathrm{m},\) and \(g \approx 10 \mathrm{m} / \mathrm{s}^{2} ?(\mathrm{c})\) Within what interval of time \((\pm \Delta t)\) must the pilot drop the bundle if it is to land within \(\pm 10\) m of the raft?

A student kicks a frictionless puck with initial speed \(v_{\mathrm{o}},\) so that it slides straight up a plane that is inclined at an angle \(\theta\) 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 \(\theta\) 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 \(O\) so that it slides across the board with initial velocity \(\left(v_{\mathrm{ox}}, v_{\mathrm{oy}}, 0\right) .\) 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 \(O\) when it does so.

A ball is thrown with initial speed \(v_{\mathrm{o}}\) up an inclined plane. The plane is inclined at an angle \(\phi\) above the horizontal, and the ball's initial velocity is at an angle \(\theta\) 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 \(R=2 v_{\mathrm{o}}^{2} \sin \theta \cos (\theta+\phi) /\left(g \cos ^{2} \phi\right)\) from its launch point. Show that for given \(v_{\mathrm{o}}\) and \(\phi,\) the maximum possible range up the inclined plane is \(R_{\max }=v_{\mathrm{o}}^{2} /[g(1+\sin \phi)]\)

A cannon shoots a ball at an angle \(\theta\) 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 \(\theta\) if \(r(t)\) is to increase throughout the ball's flight? [Hint: Using your solution to part (a) you can write down \(r^{2}\) as \(x^{2}+y^{2},\) and then find the condition that \(r^{2}\) 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 \(\omega\) Write down Newton's second law (1.48) in polar coordinates and find the tension in the string.

Verify by direct substitution that the function \(\phi(t)=A \sin (\omega t)+B \cos (\omega t)\) of (1.56) is a solution of the second-order differential equation \((1.55), \ddot{\phi}=-\omega^{2} \phi .\) (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 \(\mathbf{v}(t)\) is any vector that depends on time (for example the velocity of a moving particle) but which has constant magnitude, then \(\dot{\mathbf{v}}(t)\) is orthogonal to \(\mathbf{v}(t) .\) Prove the converse that if \(\dot{\mathbf{v}}(t)\) is orthogonal to \(\mathbf{v}(t),\) then \(|\mathbf{v}(t)|\) is constant. [Hint: Consider the derivative of \(\mathbf{v}^{2}\).] This is a very handy result. It explains why, in two- dimensional polars, \(d \hat{\mathbf{r}} / d t\) has to be in the direction of \(\hat{\boldsymbol{\phi}}\) 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.

Find expressions for the unit vectors \(\hat{\boldsymbol{\rho}}, \hat{\boldsymbol{\phi}},\) and \(\hat{\mathbf{z}}\) of cylindrical polar coordinates (Problem 1.47) in terms of the Cartesian \(\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}\). Differentiate these expressions with respect to time to find \(d \hat{\boldsymbol{\rho}} / d t, d \hat{\boldsymbol{\phi}} / d t,\) and \(d \hat{\mathbf{z}} / d t\)

Imagine two concentric cylinders, centered on the vertical \(z\) axis, with radii \(R \pm \epsilon,\) where \(\epsilon\) is very small. A small frictionless puck of thickness \(2 \epsilon\) 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 \((\rho, \phi, z)\) for its position (Problem 1.47 ), then \(\rho\) is fixed at \(\rho=R,\) while \(\phi\) 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 \(\phi_{\mathrm{o}}=20\) degrees, using the values \(R=5 \mathrm{m}\) and \(g=9.8 \mathrm{m} / \mathrm{s}^{2} .\) Make a plot of \(\phi\) against time for two or three periods. ( \(\mathbf{b}\) ) On the same picture, plot the approximate solution (1.57) with the same \(\phi_{\mathrm{o}}=20^{\circ}\) 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.