Solution: If there were a great migration of people toward

If there were a great migration of people toward the Earths equator, would the length of the day (a) get longer because of conservation of angular momentum; (b) get shorter because of conservation of angular momentum; (c) get shorter because of conservation of energy; (d) get longer because of conservation of energy; or (e) remain unaffected?

Can the diver of Fig. 11-2 do a somersault without having any initial rotation when she leaves the board?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

When a motorcyclist leaves the ground on a jump and leaves the throttle on (so the rear wheel spins), why does the front of the cycle rise up?

Suppose you are standing on the edge of a large freely rotating turntable. What happens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 11-28). Explain

If all the components of the vectors Vj and V2 were reversed in direction, how would this alter Vj X V2 ?

Name the four different conditions that could make Yl X V2 = 0.

A force F = Fj is applied to an object at a position r = x * + Xj + where the origin is at the cm. Does the torque about the cm depend on x l On y l On z1

A particle moves with constant speed along a straight line. How does its angular momentum, calculated about any point not on its path, change in time?

If the net force on a system is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zero? Give examples.

Explain how a child pumps on a swing to make it go higher.

Describe the torque needed if the person in Fig. 11-17 is to tilt the axle of the rotating wheel directly upward without it swerving to the side.

An astronaut floats freely in a weightless environment. Describe how the astronaut can move her limbs so as to (a) turn her body upside down and (b) turn her body about-face.

On the basis of the law of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate in order to keep the helicopter stable.

A wheel is rotating freely about a vertical axis with constant angular velocity. Small parts of the wheel come loose and fly off. How does this affect the rotational speed of the wheel? Is angular momentum conserved? Is kinetic energy conserved? Explain.

Consider the following vector quantities: displacement, velocity, acceleration, momentum, angular momentum, torque, (a) Which of these are independent of the choice of origin of coordinates? (Consider different points as origin which are at rest with respect to each other.) (b) Which are independent of the velocity of the coordinate system?

How does a car make a right turn? Where does the torque come from that is needed to change the angular momentum?

The axis of the Earth precesses with a period of about 25,000 years. This is much like the precession of a top. Explain how the Earths equatorial bulge gives rise to a torque exerted by the Sun and Moon on the Earth; see Fig. 11-29, which is drawn for the winter solstice (December 21). About what axis would you expect the Earths rotation axis to precess as a result of the torque due to the Sun? Does 123^/ the torque exist three months later? Explain

Why is it that at most locations on the Earth, a plumb bob does not hang precisely in the direction of the Earths center?

In a rotating frame of reference, Newtons first and second laws remain useful if we assume that a pseudoforce equal to mo)2r is acting. What effect does this assumption have on the validity of Newtons third law?

In the battle of the Falkland Islands in 1914, the shots of British gunners initially fell wide of their marks because their calculations were based on naval battles fought in the Northern Hemisphere. The Falklands are in the Southern Hemisphere. Explain the origin of their problem.

(II) What is the angle 6 between two vectors A and B, if |A X B| = A B?

(II) A particle is located at r = (4.0i + 3.5j + 6.0k) m. A force F = (9.0j - 4.0k) N acts on it. What is the torque, calculated about the origin?

(II) Consider a particle of a rigid object rotating about a fixed axis. Show that the tangential and radial vector components of the linear acceleration are: atan = a X r and aR = w X v.

(II) (a) Show that the cross product of two vectors, A = A x i + A y j + A z k, and B = Bx i + By\ + Bz k is A X B = (A y Bz - A z By) i + (A ZBX - A XBZ)j + (A x By - A y Bx)k. (b) Then show that the cross product can be written where we use the rules for evaluating a determinant. (Note, however, that this is not really a determinant, but a memory aid.)

(II) An engineer estimates that under the most adverse expected weather conditions, the total force on the highway sign in Fig. 11-32 will be F = ( 2.41 - 4.1j)kN, acting at the cm. What torque does this force exert about the base O?

(II) The origin of a coordinate system is at the center of a wheel which rotates in the xy plane about its axle which is the z axis. A force F = 215 N acts in the xy plane, at a +33.0 angle to the x axis, at the point x = 28.0 cm, y = 33.5 cm.

(II) Use the result of Problem 26 to determine (a) the vector product A X B and (b) the angle between A and B if A = 5.4i 3.5j and B = 8.5i + 5.6j + 2.0k.

(Ill) Show that the velocity v of any point in an object rotating with angular velocity dt about a fixed axis can be written v = w X r where r is the position vector of the point relative to an origin O located on the axis of rotation. Can O be anywhere on the rotation axis? Will v = a> X r if O is located at a point not on the axis of rotation?

(Ill) Let A, B, and C be three vectors, which for generality we assume do not all lie in the same plane. Show that A (B X C) = B (C X A) = C (A X B).

(I) What are the x, y, and z components of the angular momentum of a particle located at r = xi + yj + zk which has momentum p = px \ + p yj + pz k?

(I) Show that the kinetic energy K of a particle of mass ra, moving in a circular path, is K = L2/2I, where L is its angular momentum and I is its moment of inertia about the center of the circle.

(I) Calculate the angular momentum of a particle of mass ra moving with constant velocity v for two cases (see Fig. 11-33): (a) about origin O, and (b) about O'.

(II) Two identical particles have equal but opposite momenta, p and p, but they are not traveling along the A x A y , same line. Show that the total angular momentum of this system does not depend on the choice of origin.

(II) Determine the angular momentum of a 75-g particle about the origin of coordinates when the particle is at x = 4.4 m, y = -6.0 m, and it has velocity v = (3.2i - 8.0k) m/s.

(II) A particle is at the position (x, y, z) = (1.0,2.0, 3.0) m. It is traveling with a vector velocity (5.0, +2.8, 3.1) m/s. Its mass is 3.8 kg. What is its vector angular momentum about the origin?

(II) An Atwood machine (Fig. 11-16) consists of two masses, raA = 7.0 kg and raB = 8.2 kg, connected by a cord that passes over a pulley free to rotate about a fixed axis. The pulley is a solid cylinder of radius R0 = 0.40 m and mass 0.80 kg. (a) Determine the acceleration a of each mass, (b) What percentage of error in a would be made if the moment of inertia of the pulley were ignored? Ignore friction in the pulley bearings.

(II) Four identical particles of mass ra are mounted at equal intervals on a thin rod of length and mass M, with one mass at each end of the rod. If the system is rotated with angular velocity <w about an axis perpendicular to the rod

(II) Two lightweight rods 24 cm in length are mounted perpendicular to an axle and at 180 to each other (Fig. 11-34). At the end of each rod is a 480-g mass. The rods are spaced 42 cm apart along the axle. The axle rotates at 4.5 rad/s. (a) What is the component of the total angular momentum along the axle? (b) What angle does the vector angular momentum make with the axle? [Hint: Remember that the vector angular momentum must be calculated about the same 480 point for both masses, which could be the cm.1

(II) Figure 11-35 shows two masses connected by a cord passing over a pulley of radius Rq and moment of inertia I. Mass Ma slides on a frictionless surface, and MB hangs freely. Determine a formula for (a) the angular momentum of the system about the pulley axis, as a function of the speed v of mass Ma or MB, and (b) the acceleration of the masses.

(Ill) A thin rod of length I and mass M rotates about a vertical axis through its center with angular velocity on. The rod makes an angle 4> with the rotation axis. Determine the magnitude and direction of L.

(Ill) Show that the total angular momentum L = 2r* X p2 of a system of particles about the origin of an inertial reference frame can be written as the sum of the angular momentum about the cm, L* (spin angular momentum), plus the angular momentum of the cm about the origin (orbital angular momentum): L = L* + rCM X Mvcm. [Hint: See the derivation of Eq. 11-9b.]

(Ill) What is the magnitude of the force F exerted by each bearing in Fig. 11-18 (Example 11-10)? The bearings are a distance d from point O. Ignore the effects of gravity.

(Ill) Suppose in Fig. 11-18 that raB = 0; that is, only one mass, raA, is actually present. If the bearings are each a distance d from O, determine the forces FA and FB at the upper and lower bearings respectively. [Hint: Choose an origindifferent than O in Fig. 11-18such that L is parallel to co. Ignore effects of gravity.]

(Ill) Suppose in Fig. 11-18 that raA = raB = 0.60 kg, rA = rB = 0.30 m, and the distance between the bearings is 0.23m. Evaluate the force that each bearing much exert.

(II) A thin rod of mass M and length i is suspended vertically from a frictionless pivot at its upper end. A mass m of putty traveling horizontally with a speed v strikes the rod at its cm and sticks there. How high does the bottom of the rod swing?

(II) A uniform stick 1.0 m long with a total mass of 270 g is pivoted at its center. A 3.0-g bullet is shot through the stick midway between the pivot and one end (Fig. 11-36). The bullet approaches at 250 m/s and leaves at 140 m/s. With what angular speed is the stick spinning after the collision?

(II) Suppose a 5.8 X 1010kg meteorite struck the Earth at the equator with a speed v = 2.2 X 104m/s, a as shown in Fig. 11-37 and remained stuck. By what factor would this affect the rotational frequency of the Earth (1 rev/day)?

(Ill) A 230-kg beam 2.7 m in length slides broadside down the ice with a speed of 18 m/s (Fig. 11-38). A 65-kg man at rest grabs one end as it goes past and hangs on as both he and the beam go spinning down the ice. Assume frictionless motion, (a) How fast does the center of mass of the system move after the collision? (b) With what angular velocity does the system rotate about its cm?

(Ill) A thin rod of mass M and length I rests on a frictionless table and is struck at a point /A from its cm by a clay ball of mass m moving at speed v (Fig. 11-39). The ball sticks to the rod. Determine the translational and rotational motion of the rod after the collision.

(Ill) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v0 and a reverse spin of angular speed o)0 (see Fig. 11-40). A kinetic friction force acts on the ball as it initially skids across the table, (a) Explain why the balls angular momentum is conserved about point O. (b) Using conservation of angular momentum, find the critical angular speed o)C such that, if co0 = eoc , kinetic friction will bring the ball to a complete (as opposed to momentary) stop, (c) If <w0 is 10% smaller than <wc , i.e., (o0 = 0.90 (oc , determine the balls cm velocity vCM when it starts to roll without slipping. (d) If o)q is 10% larger than o)C, i.e., a)0 = 1.10 eoc , determine the balls cm velocity vCM when it starts to roll without slipping. [Hint: The ball possesses two types of angular momentum, the first due to the linear speed vCM of its cm relative to point O, the second due to the spin at angular velocity co about its own cm. The balls total L about O is the sum of these two angular momenta.l

(II) A 220-g top spinning at 15 rev/s makes an angle of 25 to the vertical and precesses at a rate of 1.00 rev per 6.5 s. If its cm is 3.5 cm from its tip along its symmetry axis, what is the moment of inertia of the top?

(II) A toy gyroscope consists of a 170-g disk with a radius of 5.5 cm mounted at the center of a thin axle 21 cm long (Fig. 11-41). The gyroscope spins at 45 rev/s. One end of its axle rests on a stand and the other end precesses horizontally about the stand, (a) H ow long does it take the gyroscope to precess once around? (b) If all the dimensions of the gyroscope were doubled (radius = 11cm, axle = 42 cm), how long would it take to precess once?

(II) Suppose the solid wheel of Fig. 11-41 has a mass of 300 g and rotates at 85 rad/s; it has radius 6.0 cm and is mounted at the center of a horizontal thin axle 25 cm long. At what rate does the axle precess?

(II) If a mass equal to half the mass of the wheel in Problem 55 is placed at the free end of the axle, what will be the precession rate now? Treat the extra mass as insignificant in size.

(II) A bicycle wheel of diameter 65 cm and mass m rotates on its axle; two 20-cm-long wooden handles, one on each side of the wheel, act as the axle. You tie a rope to a small hook on the end of one of the handles, and then spin the bicycle wheel with a flick of the hand. When you release the spinning wheel, it precesses about the vertical axis defined by the rope, instead of falling to the ground (as it would if it were not spinning).

(II) If a plant is allowed to grow from seed on a rotating platform, it will grow at an angle, pointing inward. Calculate what this angle will be (put yourself in the rotating frame) in terms of g, r, and (o. Why does it grow inward rather than outward?

(Ill) Let g' be the effective acceleration of gravity at a point on the rotating Earth, equal to the vector sum of the true value g plus the effect of the rotating reference frame (mo)2r term). See Fig. 11-42. Determine the magnitude and direction of g' relative to a radial line from the center of the Earth (a) at the North Pole, (b) at a latitude of 45.0 north, and (c) at na2r the equator. Assume that g (if a) were zero) is a constant 9.80 m/s2.

(II) Suppose the man at B in Fig. 11-26 throws the ball toward the woman at A. (a) In what direction is the ball deflected as seen in the noninertial system? (b) Determine a formula for the amount of deflection and for the (Coriolis) acceleration in this case.

(II) For what directions of velocity would the Coriolis effect on an object moving at the Earths equator be zero?

(Ill) We can alter Eqs. 11-14 and 11-15 for use on Earth by considering only the component of v perpendicular to the axis of rotation. From Fig. 11-43, we see that this is v cos A for a vertically falling object, where A is the latitude of the place on the Earth. If a lead ball is dropped vertically from a 110-m-high tower in Florence, Italy (latitude = 44), how far from the base of the tower it deflected by the Coriolis force?

Ill) An ant crawls with constant speed outward along a radial spoke of a wheel rotating at constant angular velocity cd about a vertical axis. Write a vector equation for all the forces (including inertial forces) acting on the ant.

A thin string is wrapped around a cylindrical hoop of radius R and mass M. One end of the string is fixed, and the hoop is allowed to fall vertically, starting from rest, as the string unwinds, (a) Determine the angular momentum of the hoop about its cm as a function of time. (b) What is the tension in the string as function of time?

A particle of mass 1.00 kg is moving with velocity v = (7.0i + 6.0j) m/s. (a) Find the angular momentum L relative to the origin when the particle is at ? = (2.0j + 4.0k) m. (b) At position r a force of F = 4.0 Ni is applied to the particle. Find the torque relative to the origin.

A merry-go-round with a moment of inertia equal to 1260 kg-m2 and a radius of 2.5 m rotates with negligible friction at 1.70 rad/s. A child initially standing still next to the merry-go-round jumps onto the edge of the platform straight toward the axis of rotation causing the platform to slow to 1.25 rad/s. What is her mass?

Why might tall narrow SUVs and buses be prone to rollover? Consider a vehicle rounding a curve of radius R on a flat road. When just on the verge of rollover, its tires on the inside of the curve are about to leave the ground, so the friction and normal force on these two tires are zero. The total normal force on the outside tires is Fn and the total friction force is Ffr . Assume that the vehicle is not skidding. (a) Analysts define a static stability factor SSF = w/2h, where a vehicles track width w is the distance between tires on the same axle, and h is the height of the cm above the ground. Show that the critical rollover speed is w V c = 'l R g \2 h )- [Hint: Take torques about an axis through the center of mass of the SUV, parallel to its direction of motion.] (b) Determine the ratio of highway curve radii (minimum possible) for a typical passenger car with SSF = 1.40 and an SUV with SSF = 1.05 at a speed of 90 km/h.

A spherical asteroid with radius r = 123 m and mass M = 2.25 X 1010 kg rotates about an axis at four revolutions per day. A tug spaceship attaches itself to the asteroids south pole (as defined by the axis of rotation) and fires its engine, applying a force F tangentially to the asteroids surface as shown in Fig. 11-44. If F = 265 N, how long will it take the tug to rotate the asteroids axis of rotation through an angle of 10.0 by this method?

The time-dependent position of a point object which moves counterclockwise along the circumference of a circle (radius R) in the xy plane with constant speed v is given by ? = i R cos (at + j R sin a)t where the constant (o = v/R. Determine the velocity v and angular velocity o) of this object and then show that these three vectors obey the relation v = a> X r.

The position of a particle with mass m traveling on a helical path (see Fig. 11-45) is given by ? = R cos I ttz \ 1 _ . j . + Rsm 27T_z\? d J j + zk where R and d are the radius and pitch of the helix, respectively, and z has time dependence z = vz t where vz is the (constant) component of velocity in the z direction. Determine the time-dependent angular momentum L of the particle about the origin.

A boy rolls a tire along a straight level street. The tire has mass 8.0 kg, radius 0.32 m and moment of inertia about its central axis of symmetry of 0.83 kg m2. The boy pushes the tire forward away from him at a speed of 2.1 m/s and sees that the tire leans 12 to the right (Fig. 11-46). (a) How will the resultant torque affect the subsequent motion of the tire? (b) Compare the change in angular momentum caused by this torque in 0.20 s to the original magnitude of angular momentum.

A 70-kg person stands on a tiny rotating platform with arms outstretched, (a) Estimate the moment of inertia of the person using the following approximations: the body (including head and legs) is a 60-kg cylinder, 12 cm in radius and 1.70 m high; and each arm is a 5.0-kg thin rod, 60 cm long, attached to the cylinder, (b) Using the same approximations, estimate the moment of inertia when the arms are at the persons sides, (c) If one rotation takes 1.5 s when the persons arms are outstretched, what is the time for each rotation with arms at the sides? Ignore the moment of inertia of the lightweight platform, (d) Determine the change in kinetic energy when the arms are lifted from the sides to the horizontal position.

Water drives a waterwheel (or turbine) of radius R = 3.0 m as shown in Fig. 11-47. The water enters at a speed Vi = 7.0 m/s and exits from the waterwheel at a speed v2 = 3.8 m/s. (a) If 85 kg of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? (b) What is the torque the water applies to the waterwheel? (c) If the water causes the waterwheel to make one revolution every 5.5s, how much power is delivered to the wheel?

The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moons spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)

A particle of mass m uniformly accelerates as it moves counterclockwise along the circumference of a circle of radius R: r = i R cos 0 + j R sin 0 with 6 = ft>01 + \ at2, where the constants w0 and a are the initial angular velocity and angular acceleration, respectively. Determine the objects tangential acceleration atan and determine the torque acting on the object using (a) f = r X F, (b) f = Id.

A projectile with mass m is launched from the ground and follows a trajectory given by ? = (vxot)i + ^Vyot - \ g t 2^ j where vx0 and vy0 are the initial velocities in the x and y direction, respectively, and g is the acceleration due to gravity. The launch position is defined to be the origin. Determine the torque acting on the projectile about the origin using (a) f = r X F, (b) t = dL/dt.

Most of our Solar Systems mass is contained in the Sun, and the planets possess almost all of the Solar Systems angular momentum. This observation plays a key role in theories attempting to explain the formation of our Solar System. Estimate the fraction of the Solar Systems total angular momentum that is possessed by planets using a simplified model which includes only the large outer planets with the most angular momentum. The central Sun (mass 1.99 X 1030kg, radius 6.96 X 108 m) spins about its axis once every 25 days and the planets Jupiter, Saturn, Uranus, and Neptune move in nearly circular orbits around the Sun with orbital data given in the Table below. Ignore each planets spin about its own axis.

A bicyclist traveling with speed v = 9.2 m/s on a flat road is making a turn with a radius r = 12 m. The forces acting on the cyclist and cycle are the normal force (Fn) and friction force (Ffr) exerted by the road on the tires and rag, the total weight of the cyclist and cycle. Ignore the small mass of the wheels, (a) Explain carefully why the angle 0 the bicycle makes with the vertical (Fig. 11-48) must be given by tan 0 = FfT/FN if the cyclist is to maintain balance. (b) Calculate 0 for the values given. [Hint: Consider the circular translational motion of the bicycle and rider.] (c) If the coefficient of static friction between tires and road is = 0.65, what is the minimum turning radius?

Competitive ice skaters commonly perform single, double, and triple axel jumps in which they rotate l | , 2 | , and 3 \ revolutions, respectively, about a vertical axis while airborne. For all these jumps, a typical skater remains airborne for about 0.70 s. Suppose a skater leaves the ground in an open position (e.g., arms outstretched) with moment of inertia 70 and rotational frequency /o = 1.2 rev/s, maintaining this position for 0.10 s. The skater then assumes a closed position (arms brought closer) with moment of inertia I, acquiring a rotational frequency / , which is maintained for 0.50 s. Finally, the skater immediately returns to the open position for 0.10 s until landing (see Fig. 11 - 49). (a) Why is angular momentum conserved during the skaters jump? Neglect air resistance, (b) Determine the minimum rotational frequency / during the flights middle section for the skater to successfully complete a single and a triple axel, (c) Show that, according to this model, a skater must be able to reduce his or her moment of inertia in midflight by a factor of about 2 and 5 in order to complete a single and triple axel, respectively.

A radio transmission tower has a mass of 80 kg and is 12 m high. The tower is anchored to the ground by a flexible joint at its base, but it is secured by three cables 120 apart (Fig. 11-50). In an analysis of a potential failure, a mechanical engineer needs to determine the behavior of the tower if one of the cables broke. The tower would fall away from the broken cable, rotating about its base. Determine the speed of the top of the tower as a function of the rotation angle 6. Start your analysis with the rotational dynamics equation of motion dL/dt = f net. Approximate the tower as a tall thin rod.

Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing f of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrown-off mass carries off either (a) no angular momentum, or (b) its proportional share (f) of the initial angular momentum.

A baseball bat has a sweet spot where a ball can be hit with almost effortless transmission of energy. A careful analysis of baseball dynamics shows that this special spot is located at the point where an applied force would result in pure rotation of the bat about the handle grip. Determine the location of the sweet spot of the bat shown in Fig. 11-51. The linear mass density of the bat is given roughly by (0.61 + 3.3jc2) kg/m, where x is in meters measured from the end of the handle. The entire bat is 0.84 m long. The desired rotation point should be 5.0 cm from the end where the bat is held. [Hint: Where is the cm of the bat?]

(II) A uniform stick 1.00 m long with a total mass of 330 g is pivoted at its center. A 3.0-g bullet is shot through the stick a distance x from the pivot. The bullet approaches at 250 m/s and leaves at 140 m/s (Fig. 11-36). (a) Determine a formula for the angular speed of the spinning stick after the collision as a function of x. (b) Graph the angular speed as a function of x, from x = 0 to x = 0.50 m.

(Ill) Figure 11-39 shows a thin rod of mass M and length i resting on a frictionless table. The rod is struck at a distance x from its cm by a clay ball of mass m moving at speed v. The ball sticks to the rod. (a) Determine a formula for the rotational motion of the system after the collision, (b) Graph the rotational motion of the system as a function of x, from x = 0 to x = i/2, with values of M = 450 g, m = 15 g, i = 1.20 m, and v = 12 m/s. (c) Does the translational motion depend on x l Explain.