Figure 11.12 shows four forces acting on a body. In what directions are the associated

Does Earths angular velocity vector point north or south?

Figure 11.12 shows four forces acting on a body. In what directions are the associated torques about point O? About point P? F2 S F1 S F3 S F4 S O P Figure 11.12 For Thought and Discussion 2

You stand with your right arm extended horizontally to the right. Whats the direction of the gravitational torque about your shoulder?

Although it contains no parentheses, the expression A S * B S # C S is unambiguous. Why? Is the expression a vector or a scalar?

Whats the angle between two vectors if their dot product is equal to the magnitude of their cross product?

Why does a tetherball move faster as it winds up its pole?

Why do helicopters have two rotors?

A group of polar bears is standing around the edge of a slowly rotating ice floe. If the bears all walk to the center, what happens to the rotation rate?

Tornadoes in the northern hemisphere rotate counterclockwise as viewed from above. A far-fetched idea suggests that driving on the right side of the road may increase the frequency of tornadoes. Does this idea have any merit? Explain in terms of the angular momentum imparted to the air as two cars pass.

Does a particle moving at constant speed in a straight line have angular momentum about a point on the line? About a point not on the line? In either case, is its angular momentum constant?

When you turn on a high-speed power tool such as a router, the tool tends to twist in your hands. Why?

Why is it easier to balance a basketball on your finger if its spinning?

A bug, initially at rest on a stationary, frictionless turntable, walks halfway around the turntables circumference. Describe the motion of the turntable while the bug is walking and after the bug has stopped.

If you increase the rotation rate of a precessing gyroscope, will the precession rate increase or decrease?

A car is headed north at 70 km/h. Give the magnitude and direction of the angular velocity of its 62-cm-diameter wheels.

If the car of Exercise 15 makes a 90 left turn lasting 25 s, determine the average angular acceleration of the wheels.

A wheel is spinning at 45 rpm with its axis vertical. After 15 s, its spinning at 60 rpm with its axis horizontal. Find (a) the magnitude of its average angular acceleration and (b) the angle the average angular acceleration vector makes with the horizontal.

A wheel is spinning about a horizontal axis with angular speed 140 rad/s and with its angular velocity pointing east. Find the magnitude and direction of its angular velocity after an angular acceleration of 35 rad/s2 , pointing 68 west of north, is applied for 5.0 s.

A 12-N force is applied at the point x = 3 m, y = 1 m. Find the torque about the origin if the force points in (a) the x-direction, (b) the y-direction, and (c) the z-direction.

A f o r c e F S = 1.3ni + 2.7nj N i s a p p l i e d a t t h e p o i n t x = 3.0 m, y = 0 m. Find the torque about (a) the origin and (b) the point x = -1.3 m, y = 2.4 m.

When you hold your arm outstretched, its supported primarily by the deltoid muscle. Figure 11.13 shows a case in which the deltoid exerts a 67-N force at 15 to the horizontal. If the forceapplication point is 18 cm horizontally from the shoulder joint, what torque does the deltoid exert about the shoulder? 18 cm 15 F = 67 N Deltoid muscle Figure 11.13 Exercise 21

Express the units of angular momentum (a) using only the fundamental units kilogram, meter, and second; (b) in a form involving newtons; (c) in a form involving joules.

In the Olympic hammer throw, a contestant whirls a 7.3-kg steel ball on the end of a 1.2-m cable. If the contestants arms reach an additional 90 cm from his rotation axis and if the balls speed just prior to release is 27 m/s, whats the magnitude of the balls angular momentum?

A gymnast of rotational inertia 62 kg # m2 is tumbling head over heels with angular momentum 470 kg # m2 /s. Whats her angular speed?

A 640-g hoop 90 cm in diameter is rotating at 170 rpm about its central axis. Whats its angular momentum?

A 7.4-cm-diameter baseball has mass 145 g and is spinning at 2000 rpm. Treating the baseball as a uniform solid sphere, whats its angular momentum?

A potters wheel with rotational inertia 6.40 kg # m2 is spinning freely at 19.0 rpm. The potter drops a 2.70-kg lump of clay onto the wheel, where it sticks 46.0 cm from the rotation axis. Whats the wheels subsequent angular speed?

A 3.0-m-diameter merry-go-round with rotational inertia 120 kg# m2 is spinning freely at 0.50 rev/s. Four 25-kg children sit suddenly on the edge of the merry-go-round. (a) Find the new angular speed, and (b) determine the total energy lost to friction between children and merry-go-round.

A uniform, spherical cloud of interstellar gas has mass 2.0*1030 kg, has radius 1.0*1013 m, and is rotating with period 1.4*106 years. The cloud collapses to form a star 7.0*108 m in radius. Find the stars rotation period.

A skater has rotational inertia 4.2 kg# m2 with his fists held to his chest and 5.7 kg# m2 with his arms outstretched. The skater is spinning at 3.0 rev/s while holding a 2.5-kg weight in each outstretched hand; the weights are 76 cm from his rotation axis. If he pulls his hands in to his chest, so theyre essentially on his rotation axis, how fast will he be spinning? Problems

You slip a wrench over a bolt. Taking the origin at the bolt, the other end of the wrench is at x = 18 cm, y = 5.5 cm. You apply a force F S = 88ni - 23nj N to the end of the wrench. Whats the torque on the bolt?

Vector A S points 30 counterclockwise from the x-axis. Vector B S has twice the magnitude of A S . Their product A S * B S has magnitude A2 and points in the negative z-direction. Whats the direction of vector B S ?

A baseball player extends his arm straight up to catch a 145-g baseball moving horizontally at 42 m/s. Its 63 cm from the players shoulder joint to the point the ball strikes his hand, and his arm remains stiff while it rotates about the shoulder during the catch. The players hand recoils 5.00 cm horizontally while he stops the ball. What average torque does the players arm exert on the ball?

Show that A S # 1A S * B S 2 = 0 for any vectors A S and B S .

A weightlifters barbell consists of two 25-kg masses on the ends of a 15-kg rod 1.6 m long. The weightlifter holds the rod at its center and spins it at 10 rpm about an axis perpendicular to the rod. Whats the magnitude of the barbells angular momentum?

A particle of mass m moves in a straight line at constant speed v. Show that its angular momentum about a point located a perpendicular distance b from its line of motion is mvb regardless of where the particle is on the line.

Two identical 1800-kg cars are traveling in opposite directions at 83 km/h. Each cars center of mass is 3.2 m from the center of the highway (Fig. 11.14). What are the magnitude and direction of the angular momentum of the system consisting of the two cars, about a point on the centerline of the highway? 3.2 m 3.2 m Figure 11.14 Problem 37

The dot product of two vectors is half the magnitude of their cross product. Whats the angle between the two vectors?

Biomechanical engineers have developed micromechanical devices for measuring blood flow as an alternative to dye injection following angioplasty to remove arterial plaque. One experimental device consists of a 300@m@diameter, 2.0@m@thick silicon rotor inserted into blood vessels. Moving blood spins the rotor, whose rotation rate provides a measure of blood flow. This device exhibited an 800-rpm rotation rate in tests with water flows at several m/s. Treating the rotor as a disk, what was its angular momentum at 800 rpm? (Hint: Youll need to find the density of silicon.)

Figure 11.15 shows the dimensions of a 880-g wooden baseball bat whose rotational inertia about its center of mass is 0.048 kg # m2 . If the bat is swung so its far end moves at 50 m/s, find (a) its angular momentum about the pivot P and (b) the constant torque applied about P to achieve this angular momentum in 0.25 s. (Hint: Remember the parallel-axis theorem.) 43 cm 31 cm P CM Figure 11.15 Problem 40

As an automotive engineer, youre charged with redesigning a cars wheels with the goal of decreasing each wheels angular momentum by 30% for a given linear speed of the car. Other design considerations require that the wheel diameter go from 38 cm to 35 cm. If the old wheel had rotational inertia 0.32 kg # m2 , what do you specify for the new rotational inertia?

A turntable of radius 25 cm and rotational inertia 0.0154 kg # m2 is spinning freely at 22.0 rpm about its central axis, with a 19.5-g mouse on its outer edge. The mouse walks from the edge to the center. Find (a) the new rotation speed and (b) the work done by the mouse.

A 17-kg dog is standing on the edge of a stationary, frictionless turntable of rotational inertia 95 kg # m2 and radius 1.81 m. The dog walks once around the turntable. What fraction of a full circle does the dogs motion make with respect to the ground?

A physics student is standing on an initially motionless, frictionless turntable with rotational inertia 0.31 kg # m2 . Shes holding a wheel with rotational inertia 0.22 kg # m2 spinning at 130 rpm about a vertical axis, as in Fig. 11.8. When she turns the wheel upside down, student and turntable begin rotating at 70 rpm. (a) Find the students mass, considering her to be a 30-cm-diameter cylinder. (b) Neglecting the distance between the axes of the turntable and wheel, determine the work she did in turning the wheel upside down.

Youre choreographing your schools annual ice show. You call for eight 60-kg skaters to join hands and skate side by side in a line extending 12 m. The skater at one end is to stop abruptly, so the line will rotate rigidly about that skater. For safety, you dont want the fastest skater to be moving at more than 8.0 m/s, and you dont want the force on that skaters hand to exceed 300 N. What do you determine is the greatest speed the skaters can have before they execute their rotational maneuver?

Find the angle between two vectors whose dot product is twice the magnitude of their cross product.

A circular bird feeder 19 cm in radius has rotational inertia 0.12 kg # m2 . Its suspended by a thin wire and is spinning slowly at 5.6 rpm. A 140-g bird lands on the feeders rim, coming in tangent to the rim at 1.1 m/s in a direction opposite the feeders rotation. Whats the rotation rate after the bird lands?

A force F S applied at the point x = 2.0 m, y = 0 m produces a torque 4.6kn N # m about the origin. If the x-component of F S is 3.1 N, what angle does it make with the x-axis?

S h o w t h a t t h e c r o s s p r o d u c t o f t w o v e c t o r s A S = Ax i n + Ay j n + Azk n and B S = Bx i n + By j n + Bzk n is given by A S * B S = 1AyBz - AzBy2 i n + 1AzBx - AxBz2 j n + 1AxBy - AyBx2k n. (Hint: Youll need to work out cross products of all possible pairs of the unit vectors i n, j n, and k n including with themselves.)

If youre familiar with determinants, show that the cross product can be written as a determinant: A S * B S = i n j n k n Ax Ay Az Bx By Bz (Hint: See the preceding problem.)

Jumbo is back! Jumbo is the 4.8-Mg elephant from Example 9.4. This time hes standing at the outer edge of a 15-Mg turntable of radius 8.5 m, rotating with angular velocity 0.15 s-1 on frictionless bearings. Jumbo then walks to the center of the turntable. Treating Jumbo as a point mass and the turntable as a solid disk, find (a) the angular velocity of the turntable once Jumbo reaches the center and (b) the work Jumbo does in walking to the center.

An anemometer for measuring wind speeds consists of four small cups, each with mass 124 g, mounted a pair of 32.6-cm-long rods with mass 75.7 g each, as shown in Fig. 11.16. Find the angular momentum of the anemometer when its spinning at 12.4 rev/s. You can treat the cups as point masses. 32.6 cm Figure 11.16 Problem 52

A turntable has rotational inertia I and is rotating with angular speed v about a frictionless vertical axis. A wad of clay with mass m is tossed onto the turntable and sticks a distance d from the rotation axis. The clay hits horizontally with its velocity v ! at right angles to the turntables radius, and in the same direction as the turntables rotation (Fig. 11.17). Find expressions for v that will result in (a) the turntables angular speed dropping to half its initial value, (b) no change in the turntables angular speed, and (c) the angular speed doubling. v u d Figure 11.17 Problem 53.

A uniform, solid, spherical asteroid with mass 1.2*1013 kg and radius 1.0 km is rotating with period 4.3 h. A meteoroid moving in the asteroids equatorial plane crashes into the equator at 8.4 km/s. It hits at a 58 angle to the vertical and embeds itself at the surface. After the impact the asteroids rotation period is 3.9 h. Find the meteoroids mass.

About 99.9% of the solar systems total mass lies in the Sun. Using data from Appendix E, estimate what fraction of the solar systems angular momentum about its center is associated with the Sun. Where is most of the rest of the angular momentum?

Youre a civil engineer for an advanced civilization on a solid spherical planet of uniform density. Running out of room for the expanding population, the government asks you to redesign your planet to give it more surface area. You recommend reshaping the planet, without adding any material or angular momentum, into a hollow shell whose thickness is one-fifth its outer radius. How much will your design increase the surface area, and how will it change the length of the day?

In Fig. 11.18, the lower disk, of mass 440 g and radius 3.5 cm, is rotating at 180 rpm on a frictionless shaft of negligible radius. The upper disk, of mass 270 g and radius 2.3 cm, is initially not rotating. It drops freely down onto the lower disk, and frictional forces bring the two disks to a common rotational speed. Find (a) that common speed and (b) the fraction of the initial kinetic energy lost to friction. Initial Final Figure 11.18 Problem 57

A massless spring with constant k is mounted on a turntable of rotational inertia I, as shown in Fig. 11.19. The turntable is on a frictionless vertical axle, though initially its not rotating. The spring is compressed a distance x from its equilibrium, with a mass m placed against it. When the spring is released, the mass moves at right angles to a line through the turntables center, at a distance b from the center, and slides without friction across the table and off the edge. Find expressions for (a) the linear speed of the mass and (b) the rotational speed of the turntable. (Hint: Whats conserved?) v u b v Figure 11.19 Problem 58

A solid ball of mass M and radius R is spinning with angular velocity v0 about a horizontal axis. It drops vertically onto a surface where the coefficient of kinetic friction with the ball is mk (Fig. 11.20). Find expressions for (a) the final angular velocity once its achieved pure rolling motion and (b) the time it takes to achieve this motion. Initial Final v0 v Figure 11.20 Problem 59

A time-dependent torque given by t = a + b sin ct is applied to an object thats initially stationary but is free to rotate. Here a, b, and c are constants. Find an expression for the objects angular momentum as a function of time, assuming the torque is first applied at t = 0.

Consider a rapidly spinning gyroscope whose axis is precessing uniformly in a horizontal circle of radius r, as shown in Fig. 11.10. Apply t ! = dL S /dt to show that the angular speed of precession about the vertical axis through the center of the circle is mgr/L.

When a star like our Sun exhausts its fuel, thermonuclear reactions in its core cease, and it collapses to become a white dwarf. Often the star will blow off its outer layers and lose some mass before it collapses. Suppose a star with the Suns mass and radius is rotating with period 25 days and then it collapses to a white dwarf with 60% of the Suns mass and a rotation period of 131 s. Whats the radius of the white dwarf? Compare your answer with the radii of Sun and Earth.

Pulsarsthe rapidly rotating neutron stars described in Example 11.2have magnetic fields that interact with charged particles in the surrounding interstellar medium. The result is torque that causes the pulsars spin rate and therefore its angular momentum to decrease very slowly. The table below gives values for the rotation period of a given pulsar as its been observed at the same date every 5 years for two decades. The pulsars rotational inertia is known to be 1.12*1038 kg # m2 . Make a plot of the pulsars angular momentum over time, and use the associated best-fit line, along with the rotational analog of Newtons law, to find the torque acting on the pulsar. Year of observation 1995 2000 2005 2010 2015 Angular momentum (1037 kg # m2 /s) 7.844 7.831 7.816 7.799 7.787

A system has total angular momentum L S about an axis O. Show that the systems angular momentum about a parallel axis O is given by L S = L - h ! * p ! , where p ! is the systems linear momentum and h ! is a vector from O to O (see Fig. 11.21, which also shows vectors r ! i and r ! i from each axis to the systems ith mass element mi). O O ri u ri u h u mi Figure 11.21 Problem 64

Figure 11.22 shows a demonstration gyroscope, consisting of a solid disk mounted on a shaft. The disk spins about the shaft on essentially frictionless bearings. The shaft is mounted on a stand so its free to pivot both horizontally and vertically. A weight at the far end of the shaft balances the disk, so in the configuration shown theres no torque on the system. An arrowhead mounted on the disk end of the shaft indicates the direction of the disks angular velocity. If you push on the shaft between the arrowhead and the disk, pushing horizontally away from you (i.e., into the page in Fig. 11.22), the arrowhead end of the shaft will move a. away from you (i.e., into the page). b. toward you (i.e., out of the page). c. downward. d. upward.

Figure 11.22 shows a demonstration gyroscope, consisting of a solid disk mounted on a shaft. The disk spins about the shaft on essentially frictionless bearings. The shaft is mounted on a stand so its free to pivot both horizontally and vertically. A weight at the far end of the shaft balances the disk, so in the configuration shown theres no torque on the system. An arrowhead mounted on the disk end of the shaft indicates the direction of the disks angular velocity. If you push on the shaft between the arrowhead and the disk, pushing directly upward on the bottom of the shaft, the arrowhead end of the shaft will move a. away from you (i.e., into the page). b. toward you (i.e., out of the page). c. downward. d. upward.

Figure 11.22 shows a demonstration gyroscope, consisting of a solid disk mounted on a shaft. The disk spins about the shaft on essentially frictionless bearings. The shaft is mounted on a stand so its free to pivot both horizontally and vertically. A weight at the far end of the shaft balances the disk, so in the configuration shown theres no torque on the system. An arrowhead mounted on the disk end of the shaft indicates the direction of the disks angular velocity. If an additional weight is hung on the left end of the shaft, the arrowhead will a. pivot upward until the weighted end of the shaft hits the base. b. pivot downward until the arrowhead hits the base. c. precess counterclockwise when viewed from above. d. precess clockwise when viewed from above.

Figure 11.22 shows a demonstration gyroscope, consisting of a solid disk mounted on a shaft. The disk spins about the shaft on essentially frictionless bearings. The shaft is mounted on a stand so its free to pivot both horizontally and vertically. A weight at the far end of the shaft balances the disk, so in the configuration shown theres no torque on the system. An arrowhead mounted on the disk end of the shaft indicates the direction of the disks angular velocity. If the system is precessing, and only the disks rotation rate is increased, the precession rate will a. decrease. b. increase. c. stay the same. d. become zero.