Figure 4.25 shows a child's toy, which has the shape of a

By writing a b in terms of components prove that the product rule for differentiation applies to the dot product of two vectors; that is, d(a b) da b + a db. dt dt dt

Evaluate the work done P W = f F dr = f (Fxdx Fydy) 0 0 (4.100) by the two-dimensional force F = (x2, 2xy) along the three paths joining the origin to the point P = (1, 1) as shown in Figure 4.24(a) and defined as follows: (a) This path goes along the x axis to Q = (1, 0) and then straight up to P. (Divide the integral into two pieces, fic; = f g + (b) On this path y = x2, and you can replace the term dy in (4.100) by dy = 2x dx and convert the whole integral into an integral over x. (c) This path is given parametrically as x = t3, y = t2. In this case rewrite x, y, dx, and dy in (4.100) in terms oft and dt, and convert the integral into an integral over t.

Do the same as in Problem 4.2, but for the force F = (y, x) and for the three paths joining P and Q shown in Figure 4.24(b) and defined as follows: (a) This path goes straight from P = (1, 0) to the origin and then straight to Q = (0, 1). (b) This is a straight line from P to Q. (Write y as a function of x and rewrite the integral as an integral over x.) (c) This is a quarter-circle centered on the origin. (Write x and y in polar coordinates and rewrite the integral as an integral over 0.)

A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where I am holding it. Initially the particle is moving in a circle of radius with angular velocity , but I now pull the string down through the hole until a length r remains between the hole and the particle. (a) What is the particle's angular velocity now? (b) Assuming that I pull the string so slowly that we can approximate the particle's path by a (a) (b) Figure 4.24 (a) 4.2. (b) 4.3 a circle of slowly shrinking radius, calculate the work I did pulling the string. (c) Compare your answer to part (b) with the particle's gain in kinetic energy. Figure 4.24 (a) Problem 4.2 (b) Problem 4.3

a) Consider a mass m in a uniform gravitational field g, so that the force on m is mg, where g is a constant vector pointing vertically down. If the mass moves by an arbitrary path from point 1 to point 2, show that the work done by gravity is Wgray(1 > 2) = mgh where h is the vertical height gained between points 1 and 2. Use this result to prove that the force of gravity is conservative (at least in a region small enough so that g can be considered constant). (b) Show that, if we choose axes with y measured vertically up, the gravitational potential energy is U = mgy (if we choose U = 0 at the origin).

For a system of N particles subject to a uniform gravitational field g acting vertically down, prove that the total gravitational potential energy is the same as if all the mass were concentrated at the center of mass of the system; that is, MgY a where M = E ma is the total mass and R = (X, Y, Z) is the position of the CM, with the y coordinate measured vertically up. [Hint: We know from Problem 4.5 that U, =

Near to the point where I am standing on the surface of Planet X, the gravitational force on a mass m is vertically down but has magnitude my y2 where y is a constant and y is the mass's height above the horizontal ground. (a) Find the work done by gravity on a mass m moving from r1 to r2, and use your answer to show that gravity on Planet X, although most unusual, is still conservative. Find the corresponding potential energy. (b) Still on the same planet, I thread a bead on a curved, frictionless, rigid wire, which extends from ground level to a height h above the ground. Show clearly in a picture the forces on the bead when it is somewhere on the wire. (Just name the forces so it's clear what they are; don't worry about their magnitude.) Which of the forces are conservative and which are not? (c) If I release the bead from rest at a height h, how fast will it be going when it reaches the ground?

Consider a small frictionless puck perched at the top of a fixed sphere of radius R. If the puck is given a tiny nudge so that it begins to slide down, through what vertical height will it descend before it leaves the surface of the sphere? [Hint: Use conservation of energy to find the puck's speed as a function of its height, then use Newton's second law to find the normal force of the sphere on the puck. At what value of this normal force does the puck leave the sphere?]

(a) The force exerted by a one-dimensional spring, fixed at one end, is F = -kx, where x is the displacement of the other end from its equilibrium position. Assuming that this force is conservative (which it is) show that the corresponding potential energy is , if we choose U to be zero at the equilibrium position. (b) Suppose that this spring is hung vertically from the ceiling with a mass m suspended from the other end and constrained to move in the vertical direction only. Find the extension of the new equilibrium position with the suspended mass. Show that the total potential energy (spring plus gravity) has the same form if we use the coordinate y equal to the displacement measured from the new equilibrium position at (and redefine our reference point so that U = 0 at y = 0).

Find the partial derivatives with respect to x, y, and z of the following functions: (a) f (x, y, z) = axe bxy cy2, (b)g(x, y, z) = sin (axyz2), (c) h(x, y, z) = aexYlz2, where a, b, and c are constants. Remember that to evaluate a flax you differentiate with respect to x treating y and z as constants.

Find the partial derivatives with respect to x, y, and z of the following functions: (a) f (x, y, z) = aye 2byz cz2 , (b) g(x, y, z) = cos(axy2z3), (c) h(x, y, z) = ar, where a, b, and c are constants and r = \/x2 + y2 + z2. Remember that to evaluate of/ax you differentiate with respect to x treating y and z as constants.

Calculate the gradient Vf of the following functions, f (x, y, z): (a) f = x2 + z3. (b) f = ky, where k is a constant. (c) f = r = -/x2 + y2 + z2. [Hint: Use the chain rule.] (d) f = 1/ r .

Calculate the gradient Vf of the following functions, f (x, y, z): (a) f = ln(r ), (b) f = rn, (c) f = g(r), where r = ,./x2 + y2 + z2 and g(r) is some unspecified function of r. [Hint: Use the chain rule.]

Prove that if f (r) and g(r) are any two scalar functions of r, then V (fg), fVg+ gVf

For f (r) = x2 + 2y2 3z2, use the approximation (4.35) to estimate the change in f if we move from the point r = (1, 1, 1) to (1.01, 1.03, 1.05). Compare with the exact result.

If a particle's potential energy is U(r) = k(x2 + y2 + z2), ) where k is a constant, what is the force on the particle?

A charge q in a uniform electric field Eo experiences a constant force F = q Eo. (a) Show that this force is conservative and verify that the potential energy of the charge at position r is U (r) = qE0 r. (b) By doing the necessary derivatives, check that F = V U .

Use the property (4.35) of the gradient to prove the following important results: (a) The vector Vf at any point r is perpendicular to the surface of constant f through r. (Choose a small displacement dr that lies in a surface of constant f . What is df for such a displacement?) (b) The direction of Vf at any point r is the direction in which f increases fastest as we move away from r. (Choose a small displacement dr = eu, where u is a unit vector and E is fixed and small. Find the direction of u for which the corresponding df is maximum, bearing in mind that a b = ab cos 9.)

(a) Describe the surfaces defined by the equation f = const, where f = x2 + 4y2. (b) Using the results of Problem 4.18, find a unit normal to the surface f = 5 at the point (1, 1, 1). In what direction should one move from this point to maximize the rate of change of f?

Find the curl, V x F, for the following forces: (a) F = kr; (b) F = (Ax, Bye, Cz3); (c) F = (Ay2 , Bx, Cz), where A, B, C and k are constants.

Verify that the gravitational force GMini 1r2 on a point mass m at r, due to a fixed point mass M at the origin, is conservative and calculate the corresponding potential energy.

The proof in Example 4.5 (page 119) that the Coulomb force is conservative is considerably simplified if we evaluate V x F using spherical polar coordinates. Unfortunately, the expression for V x F in spherical polar coordinates is quite messy and hard to derive. However, the answer is given inside the back cover, and the proof can be found in any book on vector calculus or mathematical methods.15 Taking the expression inside the back cover on faith, prove that the Coulomb force F = y r 2 is conservative.

Which of the following forces is conservative? (a) F = k(x , 2y, 3z) where k is a constant. (b) F = k(y, x, 0). (c) F = k(y, x, 0). For those which are conservative, find the corresponding potential energy U, and verify by direct differentiation that F = V U.

An infinitely long, uniform rod of mass p, per unit length is situated on the z axis. (a) Calculate the gravitational force F on a point mass m at a distance p from the z axis. (The gravitational force between two point masses is given in Problem 4.21.) (b) Rewrite F in terms of the rectangular coordinates (x, y, z) of the point and verify that V x F = 0. (c) Show that V x F = 0 using the expression for V x F in cylindrical polar coordinates given inside the back cover. (d) Find the corresponding potential energy U.

The proof that the condition V x F = 0 guarantees the path independence of the work fi2 F dr done by F is unfortunately too lengthy to be included here. However, the following three exercises capture the main points:16 (a) Show that the path independence of fi2 F dr is equivalent to the statement that the integral fr, F dr around any closed path F is zero. (By tradition, the symbol f is used for integrals around a closed path a path that starts and stops at the same point.) [Hint: For any two points 1 and 2 and any two paths from 1 to 2, consider the work done by F going from 1 to 2 along the first path and then back to 1 along the second in the reverse direction.] (b) Stokes's theorem asserts that A, F dr = f (V x F) dA, where the integral on the right is a surface integral over a surface for which the path F is the boundary, and ii. and dA are a unit normal to the surface and an element of area. Show that Stokes's theorem implies that if V x F = 0 everywhere, then A, F dr = 0. (c) While the general proof of Stokes's theorem is beyond our scope here, the following special case is quite easy to prove (and is an important step toward the general proof): Let F denote a rectangular closed path lying in a plane perpendicular to the z direction and bounded by the lines x = B, x = B + b, y = C and y = C c. For this simple path (traced counterclockwise as seen from above), prove Stokes's theorem that F dr = f (V x F) ill d A where it = z and the integral on the right runs over the flat, rectangular area inside F. [Hint: The integral on the left contains four terms, two of which are integrals over x and two over y. If you pair them in this way, you can combine each pair into a single integral with an integrand of the form Fx(x, C c, z) Fx(x, C, z) (or a similar term with the roles of x and y exchanged). You can rewrite this integrand as an integral over y of a Fx(x, y, z)/ay (and similarly with the other term), and you're home.]

A mass m is in a uniform gravitational field, which exerts the usual force F = mg vertically down, but with g varying with time, g = g (t). Choosing axes with y measured vertically up and defining U = mgy as usual, show that F = V U as usual, but, by differentiating E = 4mv2 + U with respect to t, show that E is not conserved. Figure 4.25 Problem 4.30

Suppose that the force F(r, t) depends on the time t but still satisfies V x F = 0. It is a mathematical fact (related to Stokes's theorem as discussed in Problem 4.25) that the work integral fi2 F(r, t) dr (evaluated at any one time t) is independent of the path taken between the points 1 and 2. Use this to show that the time-dependent PE defined by (4.48), for any fixed time t, has the claimed property that F(r, t) = V U (r, t). Can you see what goes wrong with the argument leading to Equation (4.19), that is, conservation of energy?

Consider a mass on the end of a spring of force constant and constrained to move along the horizontal axis. If we place the origin at the spring’s equilibrium position, the potential energy is . At time the mass is sitting at the origin and is given a sudden kick to the right so that it moves out to a maximum displacement at and then continues to oscillate about the origin. (a) Write down the equation for conservation of energy and solve it to give the mass’s velocity in terms of the position and the total energy . (b) Show that and use this to eliminate from your expression for . Us the result of (4.58), , to find the time for the mass to move from the origin out to a position . (c) Solve the result of part (b) to give as a function of time and show that the mass executes simple harmonic motion with period .

[Computer] A mass m confined to the x axis has potential energy U = kx4 with k > 0. (a) Sketch this potential energy and qualitatively describe the motion if the mass is initially stationary at x = 0 and is given a sharp kick to the right at t = 0. (b) Use (4.58) to find the time for the mass to reach its maximum displacement xmax = A. Give your answer as an integral over x in terms of m, A, and k. Hence find the period r of oscillations of amplitude A as an integral. (c) By making a suitable change of variables in the integral, show that the period r is inversely proportional to the amplitude A. (d) The integral of part (b) cannot be evaluated in terms of elementary functions, but it can be done numerically. Find the period for the case that m = k = A = 1.

Figure 4.25 shows a child's toy, which has the shape of a cylinder mounted on top of a hemisphere. The radius of the hemisphere is R and the CM of the whole toy is at a height h above the floor. (a) Write down the gravitational potential energy when the toy is tipped to an angle 0 from the vertical. [You need to find the height of the CM as a function of 0. It helps to think first about the height of the hemisphere's center 0 as the toy tilts.] (b) For what values of R and h is the equilibrium at 0 = 0 stable? Problems for Chapter 4 155 Figure 4.26 Problem 4.34

a) Write down the total energy E of the two masses in the Atwood machine of Figure 4.15 in terms of the coordinate x and x. (b) Show (what is true for any conservative one-dimensional system) that you can obtain the equation of motion for the coordinate x by differentiating the equation E = const. Check that the equation of motion is the same as you would obtain by applying Newton's second law to each mass and eliminating the unknown tension from the two resulting equations.

Consider the bead of Figure 4.13 threaded on a curved rigid wire. The bead's position is specified by its distance s, measured along the wire from the origin. (a) Prove that the bead's speed v is just v (Write v in terms of its components, dx/dt, etc., and find its magnitude using Pythagoras's theorem.) (b) Prove that m's' = Ftang, the tangential component of the net force on the bead. (One way to do this is to take the time derivative of the equation v2 = v v. The left side should lead you to :5' and the right to Ftang.) (c) One force on the bead is the normal force N of the wire (which constrains the bead to stay on the wire). If we assume that all other forces (gravity, etc.) are conservative, then their resultant can be derived from a potential energy U. Prove that Ftang = dU Ids. This shows that one- dimensional systems of this type can be treated just like linear systems, with x replaced by s and Fx by Ftang

Computer] (a) Verify the expression (4.59) for the potential energy of the cube balanced on a cylinder in Example 4.7 (page 130). (b) Make plots of U(8) for b = 0.9r and b = 1.1r. (You may as well choose units such that r, m, and g are all equal to 1.) (c) Use your plots to confirm the findings of Example 4.7 concerning the stability of the equilibrium at 0 = 0. Are there any other equilibrium points and are they stable?

An interesting one-dimensional system is the simple pendulum, consisting of a point mass m, fixed to the end of a massless rod (length 1), whose other end is pivoted from the ceiling to let it swing freely in a vertical plane, as shown in Figure 4.26. The pendulum's position can be specified by its angle 0 from the equilibrium position. (It could equally be specified by its distance s from equilibrium indeed s = 10but the angle is a little more convenient.) (a) Prove that the pendulum's potential energy (measured from the equilibrium level) is U(0) = mgl(1 cos 0). (4.101) Write down the total energy E as a function of 0 and 0. (b) Show that by differentiating your expression for E with respect to t you can get the equation of motion for 0 and that the equation of motion is just the familiar F = /a (where F is the torque, I is the moment of inertia, and a is the angular acceleration 0). (c) Assuming that the angle 0 remains small throughout the motion, solve for 0 (t) and show that the motion is periodic with period to = 270 / g. 27 Problem 4.36 (The subscript "o" is to emphasize that this is the period for small oscillations.)

Consider the Atwood machine of Figure 4.15, but suppose that the pulley has radius R and moment of inertia I. (a) Write down the total energy of the two masses and the pulley in terms of the coordinate x and I. (Remember that the kinetic energy of a spinning wheel is i ho2.) (b) Show (what is true for any conservative one-dimensional system) that you can obtain the equation of motion for the coordinate x by differentiating the equation E = const. Check that the equation of motion is the same as you would obtain by applying Newton's second law separately to the two masses and the pulley, and then eliminating the two unknown tensions from the three resulting equations.

A metal ball (mass m) with a hole through it is threaded on a frictionless vertical rod. A massless string (length 1) attached to the ball runs over a massless, frictionless pulley and supports a block of mass M, as shown in Figure 4.27. The positions of the two masses can be specified by the one angle 9. (a) Write down the potential energy U(9). (The PE is given easily in terms of the heights shown as h and H. Eliminate these two variables in favor of 9 and the constants b and 1. Assume that the pulley and ball have negligible size.) (b) By differentiating U(0) find whether the system has an equilibrium position, and for what values of m and M equilibrium can occur. Discuss the stability of any equilibrium positions.

Computer] Figure 4.28 shows a massless wheel of radius R, mounted on a frictionless, horizontal axle. A point mass M is glued to the edge of the wheel, and a mass m hangs from a string wrapped around the perimeter of the wheel. (a) Write down the total PE of the two masses as a function of the angle 0. (b) Use this to find the values of m and M for which there are any positions of equilibrium Describe the equilibrium positions, discuss their stability, and explain your answers in terms of torques. (c) Plot U (0) for the cases that m = 0.7M and m = 0.8M, and use your graphs to describe the behavior of the system if I release it from rest at 0 = 0. (d) Find the critical value of m/M on one side of which the system oscillates and on the other side of which it does not (if released from rest at 0 = 0).

Computer] Consider the simple pendulum of Problem 4.34. You can get an expression for the pendulum's period (good for large oscillations as well as small) using the method discussed in connection with (4.57), as follows: (a) Using (4.101) for the PE, find 0 as a function of 0. Next use (4.57), in the form t = f d04, to write the time for the pendulum to travel from 0 = 0 to its maximum value (the amplitude) O. Because this time is a quarter of the period t, you can now write down the period. Show that Figure 4.28 Problem 4.37 1 dO 2 f 1 du = to 7r f = r 0 -isin2(e13/2) sin2(4)/2) 7 . ,,/1-.2,/1- A2.2' (4.103) where ro is the period (4.102) (Problem 4.34) for small oscillations and A = sin(eD/2). [To get the first expression you will need to use the trig identity for 1 cos 4) in terms of sin2(4)/2). To get the second you need to make the substitution sin(4)/2) = Au.] These integrals cannot be evaluated in terms of elementary functions. However, the second integral is a standard integral called the complete elliptic integral of the first kind, sometimes denoted K (A2), whose values are tabulated17 and are known to computer software such as Mathematica [which calls it EllipticK(A2)]. (b) If you have access to computer software that knows this function, make a plot of r/ro for amplitudes 0 < < 3 rad. Comment. What becomes of r as the amplitude of oscillation approaches 7r? Explain.

a) If you have not already done so, do Problem 4.38(a). (b) If the amplitude eD is small then so is A = sin el3/2. If the amplitude is very small, we can simply ignore the last square root in (4.103). Show that this gives the familiar result for the small-amplitude period, r = To = 2701g. (c) If the amplitude is small but not very small, we can improve on the approximation of part (b). Use the binomial expansion to give the approximation 1/-/1 A2u2 ti 1 + 1A2u2 and show that, in this approximation, (4.103) gives r = ro[l sin2(43/2)]. What percentage correction does the second term represent for an amplitude of 45? (The exact answer for elp = 45 is 1.040 ro to four significant figures.)

a) Verify the three equations (4.68) that give x, y, z in terms of the spherical polar coordinates r, 9, (b) Find expressions for r, in terms of x, y, z.

A mass m moves in a circular orbit (centered on the origin) in the field of an attractive central force with potential energy \(U=kr^n\) . Prove that the kinetic energy is given by T=nU/2 .

In one dimension, it is obvious that a force obeying Hooke's law is conservative (since F = kx depends only on the position x, and this is sufficient to guarantee that F is conservative in one dimension). Consider instead a spring that obeys Hooke's law and has one end fixed at the origin, but whose other end is free to move in all three dimensions. (The spring could be fastened to a point in the ceiling and be supporting a bouncing mass m at its other end, for instance.) Write down the force F(r) exerted by the spring in terms of its length r and its equilibrium length ro. Prove that this force is conservative. [Hints: Is the force central? Assume that the spring does not bend.]

In Section 4.8, I claimed that a force F(r) that is central and spherically symmetric is automati-cally conservative. Here are two ways to prove it: (a) Since F(r) is central and spherically symmetric, it must have the form F(r) = f (r)r. Using Cartesian coordinates, show that this implies that V x F = 0. (b) Even quicker, using the expression given inside the back cover for V x F in spherical polars, show that V x F = 0.

Problem 4.43 suggests two proofs that a central, spherically symmetric force is automatically conservative, but neither proof makes really clear why this is so. Here is a proof that is less complete but more insightful: Consider any two points A and B and two different paths ACB and ADB connecting them as shown in Figure 4.29. Path ACB goes radially out from A until it reaches the radius rB of B, and then around a sphere (center 0) to B. Path ADB goes around a sphere of radius rA until it reaches the line OB, and then radially out to B. Explain clearly why the work done by a central, spherically symmetric force F is the same along both paths. (This doesn't prove that the work is the same along any two paths from A to B. If you want you can complete the proof by showing that any path can be approximated by a series of paths moving radially in or out and paths of constant r.)

In Section 4.8, I proved that a force F(r) = f (r)i that is central and conservative is automat-ically spherically symmetric. Here is an alternative proof: Consider the two paths ACB and ADB of Figure 4.29, but with rB = rA + dr where dr is infinitesimal. Write down the work done by F(r) going around both paths, and use the fact that they must be equal to prove that the magnitude function f (r) must be the same at points A and D; that is, f (r) = f (r) and the force is spherically symmetric.

Consider an elastic collision of two particles as in Example 4.8 (page 143), but with unequal masses, m1 0 m2. Show that the angle 0 between the two outgoing velocities satisfies 9 < 7r/2 if mi > m2, but 0 > r/2 if mi < m2

Consider a head-on elastic collision between two particles. (Since the collision is head-on, the motion is confined to a single straight line and is therefore one-dimensional.) Prove that the relative velocity after the collision is equal and opposite to that before. That is, v1 v2 = (1/1 v;), where v1 and v2 are the initial velocities and vi and v'2 the corresponding final velocities.

A particle of mass m1 and speed v1 collides with a second particle of mass m2 at rest. If the collision is perfectly inelastic (the two particles lock together and move off as one) what fraction of the kinetic energy is lost in the collision? Comment on your answer for the cases that m1 <-< m2 and that M2 << M 1'

Both the Coulomb and gravitational forces lead to potential energies of the form U = y r21, where y denotes ka1q2 in the case of the Coulomb force and Gm 1m2 for gravity, and r1 and r2 are the positions of the two particles. Show in detail that V1 U is the force on particle 1 and V2 U that on particle 2.

The formalism of the potential energy of two particles depends on the claim in (4.81) that V1U(r1 r2) = V2U(r1 r2). Prove this. (Use the chain rule for differentiation. The proof in three dimensions is notationally awkward, so prove the one-dimensional result that a a ax f (xi x2) = f (xi x2) , ax2 and then convince yourself that it extends to three dimensions.)

Write out the arguments of all the potential energies of the four-particle system in (4.94). For instance U = U(r1, r2, , r4), whereas U34 = U34 (r3 - r4). Show in detail that the net force on particle 3 (for instance) is given by V3 U. [You know that the separate forces, internal and external, are given by (4.92) and (4.93).]

Consider the four-particle system of Section 4.10. (a) Write down the workKE theorem for each of the four particles separately and, by adding these four equations, show that the change in the total KE in a short time interval dt is dT = Wtot where Wtot is the total work done on all particles by all forces. [This shouldn't take more than two or three lines.] (b) Next show that Wtot = dU where dU is the change in total PE during the same time interval. Deduce that the total mechanical energy E = T U is conserved.

(a) Consider an electron (charge e and mass m) in a circular orbit of radius r around a fixed proton (charge +e). Remembering that the inward Coulomb force keg /r2 is what gives the electron its centripetal acceleration, prove that the electron's KE is equal to 4 times its PE; that is, T = 1U and hence E = z U. (This result is a consequence of the so-called virial theorem. See Problem 4.41.) Now consider the following inelastic collision of an electron with a hydrogen atom: Electron number 1 is in a circular orbit of radius r around a fixed proton. (This is the hydrogen atom.) Electron 2 approaches from afar with kinetic energy T2. When the second electron hits the atom, the first electron is knocked free, and the second is captured in a circular orbit of radius r'. (b) Write down an expression for the total energy of the three-particle system in general. (Your answer should contain five terms, three PEs but only two KEs, since the proton is considered fixed.) (c) Identify the values of all five terms and the total energy E long before the collision occurs, and again long after it is all over. What is the KE of the outgoing electron 1 once it is far away? Give your answers in terms of the variables T2, r, and r'.