A 70-kg person stands on a tiny rotating platform with

What is the angular momentum of a 0.210-kg ball

(a) What is the angular momentum of a 2.8-kg uniform

A person stands, hands at his side, on a platform that is rotating at a rate of 0.90 rev/s. If he raises his arms to a horizontal position, Fig. 11-30, the speed of rotation decreases to 0.70 rev/s. (a) Why? (b) By what factor has his moment of inertia changed? FIGURE 11-30 Problem 3.

A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of 2.5 rev/s. If her initial moment of inertia was 4.6 kg-m2, what is her final moment of inertia? How does she physically accomplish this change?

A diver (such as the one shown in Fig. 11-2) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed (rev/s) when in the straight position?

A uniform horizontal rod of mass M and length I rotates with angular velocity a) about a vertical axis through its center. Attached to each end of the rod is a small mass m. Determine the angular momentum of the system about the axis.

Determine the angular momentum of the Earth (a) about its rotation axis (assume the Earth is a uniform sphere), and (b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass = 6.0 X 1024kg and radius = 6.4 X 106m, and is 1.5 X 108km from the Sun.

(a) What is the angular momentum of a figure skater spinning at 2.8 rev/s with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 48 kg? (b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?

A person stands on a platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is /P. The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia /w and angular velocity % . What will be the angular velocity wP of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60 angle to the vertical, (c) vertically downward? (d) What will wP be if the person reaches up and stops the wheel in part (a)?

A uniform disk turns at 3.7 rev/s around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disks diameter, is dropped onto the freely spinning disk, Fig. 11-31. They then turn together around the spindle with their centers superposed. What is the angular frequency in rev/s of the \ combination?FIGURE 11-31

A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 kg m2. The platform rotates without friction with angular velocity 0.95 rad/s. The person walks radially to the edge of the platform, (a) Calculate the angular velocity when the person reaches the edge. (b) Calculate the rotational kinetic energy of the system of platform plus person before and after the persons walk.

A potters wheel is rotating around a vertical axis through its center at a frequency of 1.5 rev/s. The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 2.6-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?

A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0.80 rad/s. Its total moment of inertia is 1760 kg m2. Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now? What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity (Oq about a vertical axis through its center, and the woman begins walking with speed v (relative to the platform) toward the center of the platform, (a) Determine the angular velocity of the system as a function of time. (b) What will be the angular velocity when the woman reaches the center?

A nonrotating cylindrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed . Assuming no external torques, what is the final common angular speed of the two disks?

Suppose our Sun eventually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Suns new rotation rate be? (Take the Suns current period to be about 30 days.) What would be its final kinetic energy in terms of its initial kinetic energy of today?

Hurricanes can involve winds in excess of 120 km/h at the outer edge. Make a crude estimate of (a) the energy, and (b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 kg/m3) of radius 85 km and height 4.5 km.

An asteroid of mass 1.0 X 105kg, traveling at a speed of 35km/s relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earths rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.

Suppose a 65-kg person stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1850 kg m2. The turntable is at rest initially, but when the person begins running at a speed of 3.8 m/s (with respect to the turntable) around its edge, the turntable begins to rotate

If vector A points along the negative x axis and vector B along the positive z axis, what is the direction of (a) A X B and (ib) B X A? (c) What is the magnitude of A X B and B X A?

Show that (a) i X i = j X j = k X k = 0, ( b ) i X j = k, i X k = j, and j X k = i.

The directions of vectors A and B are given below for several cases. For each case, state the direction of A X B. (a) A points east, B points south. (b) A points east, B points straight down, (c) A points straight up, B points north. (,d) A points straight up, B points straight down.

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

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?

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.

(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.)

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? FIGURE 11-32 Problem 27.

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.

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?

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).

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?

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.

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'.FIGURE 11-33 f Problem 34.

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

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.

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?

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.

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

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 point for both masses, which could be the cm.1FIGURE 11-34 Problem 40.

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. FIGURE 11-35 Problem 41.

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.

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.]

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.

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.]

Suppose in Fig. 11-18 that raA = raB = 0.60 kg, rA = rB = 0.30 m, and the distance between the bearings is

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?

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? FIGURE 11-36 Problems 48 and 83.

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)?FIGURE 11-37 Problem 49.

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? FIGURE 11-38 Problem 50.

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.

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. FIGURE 11-40 Problem 52.

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?

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? FIGURE 11-41 Awheel, rotating about a horizontal axle supported at one end, precesses. Problems 54,55, and 56.

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?

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.

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). Estimate the rate and direction nf nreeession if the wheel rotates

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?

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.FIGURE 11-42 Problem 59.

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.

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

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?FIGURE 11-43 Problem 62. Object of mass m falling vertically to Earth at a latitude A.

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. Take the x axis along the spoke, y perpendicular to the spoke nointinp to the ants left, and the 7 axis vertieallv nnward

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.FIGURE 11-45 Problem 70.

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.FIGURE 11-46 Problem 71.

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. (e) From your answer to nart (rf\ wonlH von exnent it to be harder or easier to lift

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 revolu-Problem 73.

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. Mean Distance from Orbital Period Mass Planet S u n (x l0 6km) (Earth Years) (x 1025kg) Jupiter 778 11.9 190 Saturn 1427 29.5 56.8 Uranus 2870 84.0 8.68

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?FIGURE 11-48 Problem 78.

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.FIGURE 11-50 Problem 80.

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?FIGURE 11-51 Problem 82.

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.

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.