On a certain planet, which is perfectly spherically

Be sure you understand why a pendulum in equilibrium hanging in a car that is accelerating forward tilts backward, and then consider the following: A helium balloon is anchored by a massless string to the floor of a car that is accelerating forward with acceleration A. Explain clearly why the balloon tends to tilt forward and find its angle of tilt in equilibrium. [Hint: Helium balloons float because of the buoyant Archimedean force, which results from a pressure gradient in the air. What is the relation between the directions of the gravitational field and the buoyant force?]

A donut-shaped space station (outer radius R) arranges for artificial gravity by spinning on the axis of the donut with angular velocity co. Sketch the forces on, and accelerations of, an astronaut standing in the station (a) as seen from an inertial frame outside the station and (b) as seen in the astronaut's personal rest frame (which has a centripetal acceleration A = to2R as seen in the inertial frame). What angular velocity is needed if R = 40 meters and the apparent gravity is to equal the usual value of about 10 m/s2? (c) What is the percentage difference between the perceived g at a six-foot astronaut's feet (R = 40 m) and at his head (R = 38 m)?

(a) Consider the tidal force (9.12) on a mass m at the position P of Figure 9.4. Write d as (do Re) = do(1 Re/ do) and use the binomial approximation (1 c)-2 1 + 2E to show that Fria (2G Minm Rel do3) X. Confirm the direction of the force shown in Figure 9.4 and make a numerical comparison of the tidal force with the gravitational force mg of the earth. (b) Do the corresponding calculations for the force at the point R. Compare this force with that of part (a) (magnitude and direction).

Do the same calculations as in Problem 9.3(a) but for the tidal force at the point Q in Figure 9.4. [In this case write d/d2 = d/d3 and use the binomial approximation in the form (1 + c)-3 = 1 3E.]

Review the derivation of the tidal potential energy (9.16) of a drop of water at the point Q in Figure 9.5 and then give in detail the derivation of (9.17) for the tidal PE at the point P.

Let h (8) denote the height of the ocean at any point T on the surface, where h (8) is measured up from the level at the point Q of Figure 9.5 and 6 is the polar angle TOR of T . Given that the surface of the ocean is an equipotential, show that h(8) = ho cost 8, where ho = 3 Mn, R:/ (2 Me clo3). Sketch and describe the shape of the ocean's surface, bearing in mind that ho << Re. [Hint: You will need to evaluate Utid ( T ) as given by (9.13), with d equal to the distance MT . To do this you need to find d by the law of cosines and then approximate d-1 using the binomial approximation, being very careful to keep all terms through order (Re/do)2. Neglect any effects of the sun.]

(a) Explain the relation (9.30) between the derivatives of a vector Q in two frames So and S for the special case that Q is fixed in the frame S. (b) Do the same for a vector Q that is fixed in the frame So and compare with your answer to part (a).

What are the directions of the centrifugal and Coriolis forces on a person moving (a) south near the North Pole, (b) east on the equator, and (c) south across the equator?

A bullet of mass m is fired with muzzle speed vo horizontally and due north from a position at colatitude 8. Find the direction and magnitude of the Coriolis force in terms of m, vo, 8, and the earth's angular velocity Q. How does the Coriolis force compare with the bullet's weight if vo = 1000 m/s and 0 = 40 deg?

The derivation of the equation of motion (9.34) for a rotating frame made the assumption that the angular velocity St was constant. Show that if St 0 then there is a third "fictitious force," sometimes called the azimuthal force, on the right side of (9.34) equal to mr x

In this problem you will prove the equation of motion (9.34) for a rotating frame using the Lagrangian approach. As usual, the Lagrangian method is in many ways easier than the Newtonian (except that it calls for some slightly tricky vector gymnastics), but is perhaps less insightful. Let S be a noninertial frame rotating with constant angular velocity St relative to the inertial frame So. Let both frames have the same origin, 0 = 0'. (a) Find the Lagrangian L = T U in terms of the coordinates r and r of S. [Remember that you must first evaluate T in the inertial frame. In this connection, recall that vo = v + St x r.] (b) Show that the three Lagrange equations reproduce (9.34) precisely.

a) Show that to design a static structure in a rotating frame (such as a space station) one can use the ordinary rules of statics except that one must include the extra "fictitious" centrifugal force. (b) I wish to place a puck on a rotating horizontal turntable (angular velocity Q) and to have it remain at rest on the table, held by the force of static friction (coefficient p,). What is the maximum distance from the axis of rotation at which I can do this? (Argue from the point of view of an observer in the rotating frame.)

Show that the angle a between a plumb line and the direction of the earth's center is well approximated by tan a = (R,Q2 sin 26)/(2g), where g is the observed free-fall acceleration and we assume the earth is perfectly spherically symmetric. Estimate the maximum and minimum values of the magnitude of a.

I am spinning a bucket of water about its vertical axis with angular velocity C2. Show that, once the water has settled in equilibrium (relative to the bucket), its surface will be a parabola. (Use cylindrical polar coordinates and remember that the surface is an equipotential under the combined effects of the gravitational and centrifugal forces.)

On a certain planet, which is perfectly spherically symmetric, the free-fall acceleration has magnitude g = go at the North Pole and g = Ago at the equator (with 0 < A < 1). Find g(9), the free-fall acceleration at colatitude 9 as a function of 9.

The center of a long frictionless rod is pivoted at the origin and the rod is forced to rotate at a constant angular velocity Q in a horizontal plane. Write down the equation of motion for a bead that is threaded on the rod, using the coordinates x and y of a frame that rotates with the rod (with x along the rod and y perpendicular to it). Solve for x (t). What is the role of the centrifugal force? What of the Coriolis force?

Consider the bead threaded on a circular hoop of Example 7.6 (page 260), working in a frame that rotates with the hoop. Find the equation of motion of the bead, and check that your result agrees with Equation (7.69). Using a free-body diagram, explain the result (7.71) for the equilibrium positions.

A particle of mass m is confined to move, without friction, in a vertical plane, with axes x horizontal and y vertically up. The plane is forced to rotate with constant angular velocity Q about the y axis. Find the equations of motion for x and y, solve them, and describe the possible motions.

I am standing (wearing crampons) on a perfectly frictionless flat merry-go-round, which is rotating counterclockwise with angular velocity Q about its vertical axis. (a) I am holding a puck at rest just above the floor (of the merry-go-round) and release it. Describe the puck's path as seen from above by an observer who is looking down from a nearby tower (fixed to the ground) and also as seen by me on the merry-go-round. In the latter case explain what I see in terms of the centrifugal and Coriolis forces. (b) Answer the same questions for a puck which is released from rest by a long-armed spectator who is standing on the ground leaning over the merry-go-round.

Consider a frictionless puck on a horizontal turntable that is rotating counterclockwise with angular velocity Q. (a) Write down Newton's second law for the coordinates x and y of the puck as seen by me standing on the turntable. (Be sure to include the centrifugal and Coriolis forces, but ignore the earth's rotation.) (b) Solve the two equations by the trick of writing ri = x iy and guessing a solution of the form /I = e-`at. [In this case as in the case of critically damped SHM discussed in Section 5.4 you get only one solution this way. The other has the same form (5.43) we found for the second solution in damped SHM.] Write down the general solution. (c) At time t = 0, I push the puck from position ro = (x0, 0) with velocity v, = (vxo, vyo) (all as measured by me on the turntable). Show that x(t) = (xo vxot) cos Qt (vyo Cbco)t sin Qt y(t) = (x0 + v xot) sin Qt + (v,,0 + Qxo)t cos Qt I (9.72) (d) Describe and sketch the behavior of the puck for large values of t. [Hint: When t is large the terms proportional to t dominate (except in the case that both their coefficients are zero). With t large, write (9.72) in the form x(t) = t (Bi cos Qt B2 sin Qt), with a similar expression for y(t), and use the trick of (5.11) to combine the sine and cosine into a single cosine or sine, in the case of y(t). By now you can recognize that the path is the same kind of spiral, whatever the initial conditions (with the one exception mentioned).]

When a puck slides on a rotating turntable, as in Problems 9.20 and 9.24, it can come instantaneously to rest. Sketch the shape of the path when this happens and explain. If you did Problem 9.24, comment on the relevance of this result to part (d) of that problem.

If a negative charge q (an electron, for example) in an elliptical orbit around a fixed positive charge Q is subjected to a weak uniform magnetic field B, the effect of B is to make the ellipse precess slowly an effect known as Larmor precession. To prove this, write down the equation of motion of the negative charge in the field of Q and B. Now rewrite it for a frame rotating with angular velocity Si [Remember that this changes both d2r/dt2 and drldt.] Show that by suitable choice of St you can arrange that the terms involvingr cancel out, but that you are left with one term involving B x (B x r). If B is weak enough this term can certainly be neglected. Show that in this case the orbit in the rotating frame is an ellipse (or hyperbola). Describe the appearance of the ellipse as seen in the original nonrotating frame.

Here is an unusual way to solve the two-dimensional isotropic oscillator the motion of a particle subject to a force kr. Show that by choosing a suitable rotating reference frame, you can arrange that the centrifugal force exactly cancels the force kr. Recalling the analogy between the Coriolis and magnetic forces, you should be able to write down the general solution for the motion as seen in the rotating frame. If you write your solution in the complex form of Section 2.7, then you can transform back to the nonrotating frame by multiplying by a suitable rotating complex number. Show that the general solution is an ellipse. [See Problem 8.11 for some guidance on this last part.]

[Computer] Use a suitable plotting program (such as ParametricPlot in Mathematica) to plot the orbits (9.72) of the puck of Problem 9.20 on a rotating turntable with xo = Q = 1 and the following initial velocities vo: (a) (0, 1), (b) (0, 0), (c) (0, 1), (d) (-0.5, 0.5), (e) (-0.7, 0.7), (f) (0, 0.1). Comment on any interesting features.

A high-speed train is traveling at a constant 150 m/s (about 300 mph) on a straight, horizontal track across the South Pole. Find the angle between a plumb line suspended from the ceiling inside the train and another inside a but on the ground. In what direction is the plumb line on the train deflected?

In Section 9.8, we used a method of successive aproximations to find the orbit of an object that is dropped from rest, correct to first order in the earth's angular velocity Q. Show in the same way that if an object is thrown with initial velocity vo from a point 0 on the earth's surface at colatitude 0, then to first order in Q its orbit is x = vxot Q (vvo cos 0 vzo sin 6)t2 4C2gt3 sin 0 y = vyot Q (vxo cos 0)t2 (9.73) z = vzot Zgt2+ (vxo sin 0)t2. [First solve the equations of motion (9.53) in zeroth order, that is, ignoring Q entirely. Substitute your zeroth-order solution for and z into the right side of equations (9.53) and integrate to give the next approximation. Assume that vo is small enough that air resistance is negligible and that g is a constant throughout the flight.]

In Section 9.8, we discussed the path of an object that is dropped from a very tall stepladder above the equator. (a) Sketch this path as seen from a tower to the north of the drop and fixed to the earth. Explain why the object lands to the east of its point of release. (b) Sketch the same experiment as seen by an inertial observer floating in space to the north of the drop. Explain clearly (from this point of view) why the object lands to the east of its point of release. [Hint: The object's angular momentum about the earth's center is conserved. This means that the object's angular velocity cb changes as it falls.]

Use the result (9.73) of Problem 9.26 to do the following: A naval gun shoots a shell at colatitude 0 in a direction that is a above the horizontal and due east, with muzzle speed vo. (a) Ignoring the earth's rotation (and air resistance), find how long (t) the shell would be in the air and how far away (R) it would land. If vo = 500 m/s and a = 20, what are t and R? (b) A naval gunner spots an enemy ship due east at the range R of part (a) and, forgetting about the Coriolis effect, aims his gun exactly as in part (a). Find by how far north or south, and in which direction, the shell will miss the target, in terms of Q, vo, a, 0, and g. (It will also miss in the eastwest direction but this is perhaps less critical.) If the incident occurs at latitude 50 north (0 = 40), what is this distance? What if the latitude is 50 south? This problem is a serious issue in long-range gunnery: In a battle near the Falkland Islands in World War I, the British navy consistently missed German ships by many tens of yards because they apparently forgot that the Coriolis effect in the southern hemisphere is opposite to that in the north.

(a) A baseball is thrown vertically up with initial speed vo from a point on the ground at colatitude 0. Use the solution (9.73) of Problem 9.26 to show that the ball will return to the ground a distance (4Q vo3 sin 0) / (3g2) to the west of its launch point. (b) Estimate the size of this effect on the equator if vo = 40 m/s. (c) Sketch the ball's orbit as seen from the north (by an observer fixed to the earth). Compare with the orbit of a ball dropped from a point above the equator, and explain why the Coriolis effect moves the dropped ball to the east, but the thrown ball to the west.

The Coriolis force can produce a torque on a spinning object. To illustrate this, consider a horizontal hoop of mass m and radius r spinning with angular velocity w about its vertical axis at colatitude 0. Show that the Coriolis force due to the earth's rotation produces a torque of magnitude mw 2r2 sin 0 directed to the west, where Q is the earth's angular velocity. This torque is the basis of the gyrocompass.

The Compton generator is a beautiful demonstration of the Coriolis force due to the earth's rotation, invented by the American physicist A. H. Compton (1892-1962, best known as author of the Compton effect) while he was still an undergraduate. A narrow glass tube in the shape of a torus or ring (radius R of the ring >> radius of the tube) is filled with water, plus some dust particles to let one see any motion of the water. The ring and water are initially stationary and horizontal, but the ring is then spun through 180about its eastwest diameter. Explain why this should cause the water to move around the tube. Show that the speed of the water just after the 180 turn should be 2QR cos 0, where Q is the earth's angular velocity, and 0 is the colatitude of the experiment. What would this speed be if R = 1 m and 19 = 40? Compton measured this speed with a microscope and got agreement within 3%.

Do all parts of Problem 9.28, but find the distance by which the shell misses its target in both the northsouth and eastwest directions. [Hint: In this case you must recognize that the time of flight is affected by the Coriolis effect.]

The general solution for the small-amplitude motion of a Foucault pendulum is given by (9.66). If at t = 0 the pendulum is at rest with x = A and y = 0, find the two coefficients C1 and C2, and show that because Q << coo they are well approximated as C1 = C2 = A/2, giving the solution (9.67).

At a point P on the earth's surface, an enormous perfectly flat and frictionless platform is built. The platform is exactly horizontal that is, perpendicular to the local free-fall acceleration gp. Find the equation of motion for a puck sliding on the platform and show that it has the same form as (9.61) for the Foucault pendulum except that the pendulum's length L is replaced by the earth's radius R. What is the frequency of the puck's oscillations and what is that of its Foucault precession? [Hints: Write the puck's position vector, relative to the earth's center 0 as R r, where R is the position of the point P and r = (x, y, 0) is the puck's position relative to P. The contribution to the centrifugal force involving R can be absorbed into gp and the contribution involving r is negligible. The restoring force comes from the variation of g as the puck moves.] To check the validity of your approximations, compare the approximate size of the gravitational restoring force, the Coriolis force, and the neglected term m (St x r) x St in the centrifugal force.