Chapter 11

A car travels at \(80 \mathrm{~km} / \mathrm{h}\) on a level road in the positive direction of an \(x\) axis. Each tire has a diameter of \(66 \mathrm{~cm} .\) Relative to a woman riding in the car and in unit-vector notation, what are the velocity \(\vec{v}\) at the (a) center, (b) top, and (c) bottom of the tire and the magnitude \(a\) of the acceleration at the (d) center, (e) top, and (f) bottom of each tire? Relative to a hitchhiker sitting next to the road and in unit-vector notation, what are the velocity \(\vec{v}\) at the (g) center, (h) top, and (i) bottom of the tire and the magnitude \(a\) of the acceleration at the (j) center, (k) top, and (I) bottom of each tire?

An automobile traveling at \(80.0 \mathrm{~km} / \mathrm{h}\) has tires of \(75.0 \mathrm{~cm}\) diameter. (a) What is the angular speed of the tires about their axles? (b) If the car is brought to a stop uniformly in 30.0 complete turns of the tires (without skidding), what is the magnitude of the angular acceleration of the wheels? (c) How far does the car move during the braking?

A \(140 \mathrm{~kg}\) hoop rolls along a horizontal floor so that the hoop's center of mass has a speed of \(0.150 \mathrm{~m} / \mathrm{s}\). How much work must be done on the hoop to stop it?

A uniform solid sphere rolls down an incline. (a) What must be the incline angle if the linear acceleration of the center of the sphere is to have a magnitude of \(0.10 \mathrm{~g} ?\) (b) If a frictionless block were to slide down the incline at that angle, would its acceleration magnitude be more than, less than, or equal to \(0.10 \mathrm{~g}\) ? Why?

A \(1000 \mathrm{~kg}\) car has four \(10 \mathrm{~kg}\) wheels. When the car is moving, what fraction of its total kinetic energy is due to rotation of the wheels about their axles? Assume that the wheels are uniform disks of the same mass and size. Why do you not need to know the radius of the wheels?

A solid cylinder of radius \(10 \mathrm{~cm}\) and mass \(12 \mathrm{~kg}\) starts from rest and rolls without slipping a distance \(L=6.0 \mathrm{~m}\) down a roof that is inclined at angle \(\theta=30^{\circ}\) (a) What is the angular speed of the cylinder about its center as it leaves the roof? (b) The roof's edge is at height \(H=5.0 \mathrm{~m}\). How far horizontally from the roof's edge does the cylinder hit the level ground?

A constant horizontal force \(\vec{F}_{\text {app }}\) of magnitude \(10 \mathrm{~N}\) is applied to a wheel of mass \(10 \mathrm{~kg}\) and radius \(0.30 \mathrm{~m}\). The wheel rolls smoothly on the horizontal surface, and the acceleration of its center of mass has magnitude \(0.60 \mathrm{~m} / \mathrm{s}^{2}\). (a) In unit-vector notation, what is the frictional force on the wheel? (b) What is the rotational inertia of the wheel about the rotation axis through its center of mass?

A solid brass ball of mass 0.280 g will roll smoothly along a loop-the-loop track when released from rest along the straight section. The circular loop has radius \(R=14.0 \mathrm{~cm},\) and the ball has radius \(r

A ball of mass \(M\) and radius \(R\) rolls smoothly from rest down a ramp and onto a circular loop of radius \(0.48 \mathrm{~m}\). The initial height of the ball is \(h=0.36 \mathrm{~m}\). At the loop bottom, the magnitude of the normal force on the ball is \(2.00 M g\). The ball consists of an outer spherical shell (of a certain uniform density) that is glued to a central sphere (of a different uniform density). The rotational inertia of the ball can be expressed in the general form \(I=\beta M R^{2},\) but \(\beta\) is not 0.4 as it is for a ball of uniform density. Determine \(\bar{\beta}\).

A bowler throws a bowling ball of radius \(R=11 \mathrm{~cm}\) along a lane. The ball (Fig. \(11-38\) ) slides on the lane with initial speed \(v_{\mathrm{com}}=8.5 \mathrm{~m} / \mathrm{s}\) and initial angular speed \(\omega_{0}=0 .\) The coefficient of kinetic friction between the ball and the lane is \(0.21 .\) The kinetic frictional force \(\vec{f}_{k}\) acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When speed \(v_{\text {com }}\) has decreased enough and angular speed \(\omega\) has increased enough, the ball stops sliding and then rolls smoothly. (a) What then is \(v_{\operatorname{com}}\) in terms of \(\omega ?\) During the sliding, what are the ball's (b) linear acceleration and (c) angular acceleration? (d) How long does the ball slide? (e) How far does the ball slide? (f) What is the linear speed of the ball when smooth rolling begins?

A Cylindrical object of mass \(M\) and radius \(R\) rolls smoothly from rest down a ramp and onto a horizontal section. From there it rolls off the ramp and onto the floor, landing a horizontal distance \(d=0.506 \mathrm{~m}\) from the end of the ramp. The initial height of the object is \(H=0.90 \mathrm{~m}\) the end of the ramp is at height \(h=0.10 \mathrm{~m}\). The object consists of an outer cylindrical shell (of a certain uniform density) that is glued to a central cylinder (of a different uniform density). The rotational inertia of the object can be expressed in the general form \(I=\beta M R^{2},\) but \(\beta\) is not 0.5 as it is for a cylinder of uniform density. Determine \(\beta\)

A yo-yo has a rotational inertia of \(950 \mathrm{~g} \cdot \mathrm{cm}^{2}\) and a mass of \(120 \mathrm{~g}\). Its axle radius is \(3.2 \mathrm{~mm},\) and its string is \(120 \mathrm{~cm}\) long. The yo-yo rolls from rest down to the end of the string. (a) What is the magnitude of its linear acceleration? (b) How long does it take to reach the end of the string? As it reaches the end of the string, what are its (c) linear speed, (d) translational kinetic energy, (e) rotational kinetic energy, and (f) angular speed?

In \(1980,\) over San Francisco Bav, a large yo-yo was released from a crane. The \(116 \mathrm{~kg}\) yo-yo consisted of two uniform disks of radius \(32 \mathrm{~cm}\) connected by an axle of radius \(3.2 \mathrm{~cm}\). What was the magnitude of the acceleration of the yo-yo during (a) its fall and (b) its rise? (c) What was the tension in the cord on which it rolled? (d) Was that tension near the cord's limit of \(52 \mathrm{kN} ?\) Suppose you build a scaled-up version of the yo-yo (same shape and materials but larger). (e) Will the magnitude of your yo-yo's acceleration as it falls be greater than, less than, or the same as that of the San Francisco yo-yo? (f) How about the tension in the cord?

In unit-vector notation, what is the net torque about the origin on a flea located at coordinates \((0,-4.0 \mathrm{~m}, 5.0 \mathrm{~m})\) when forces \(\vec{F}_{1}=(3.0 \mathrm{~N}) \mathrm{k}\) and \(\vec{F}_{2}=(-2.0 \mathrm{~N}) \mathrm{j}\) act on the flea?

A plum is located at coordinates \((-2.0 \mathrm{~m}, 0,4.0 \mathrm{~m}) .\) In unit-vector notation, what is the torque about the origin on the plum if that torque is due to a force \(F\) whose only component is (a) \(F_{x}=6.0 \mathrm{~N},\) (b) \(F_{x}=-6.0 \mathrm{~N},\) (c) \(F_{z}=6.0 \mathrm{~N},\) and (d) \(F_{z}=-6.0 \mathrm{~N} ?\)

In unit-vector notation, what is the torque about the origin on a particle located at coordinates \((0,-4.0 \mathrm{~m}, 3.0 \mathrm{~m})\) if that torque is due to (a) force \(\vec{F}_{1}\) with components \(F_{1 x}=2.0 \mathrm{~N}, F_{1 y}=F_{1 z}=0,\) and (b) force \(\vec{F}_{2}\) with components \(F_{2 x}=0, F_{2 y}=2.0 \mathrm{~N}, F_{2 t}=4.0 \mathrm{~N} ?\)

Force \(\vec{F}=(2.0 \mathrm{~N}) \mathrm{i}-(3.0 \mathrm{~N}) \mathrm{k}\) acts on a pebble with position vector \(\vec{r}=(0.50 \mathrm{~m}) \mathrm{j}-(2.0 \mathrm{~m}) \hat{\mathrm{k}}\) relative to the origin. In unitvector notation, what is the resulting torque on the pebble about (a) the origin and (b) the point \((2.0 \mathrm{~m}, 0,-3.0 \mathrm{~m}) ?\)

In unit-vector notation, what is the torque about the origin on a jar of jalapeno peppers located at coordinates \((3.0 \mathrm{~m},-2.0 \mathrm{~m}\), \(4.0 \mathrm{~m})\) due to (a) force \(\vec{F}_{1}=(3.0 \mathrm{~N}) \mathrm{i}-(4.0 \mathrm{~N}) \mathrm{j}+(5.0 \mathrm{~N}) \mathrm{k}\) (b) force \(\vec{F}_{2}=(-3.0 \mathrm{~N}) \hat{\mathrm{i}}-(4.0 \mathrm{~N}) \mathrm{j}-(5.0 \mathrm{~N}) \mathrm{k},\) and (c) the vector sum of \(\vec{F}_{1}\) and \(\vec{F}_{2} ?\) (d) Repeat part (c) for the torque about the point with coordinates \((3.0 \mathrm{~m}, 2.0 \mathrm{~m}, 4.0 \mathrm{~m})\).

Force \(\vec{F}=(-8.0 \mathrm{~N}) \hat{\mathrm{i}}+(6.0 \mathrm{~N}) \mathrm{j}\) acts on a particle with position vector \(\vec{r}=(3.0 \mathrm{~m}) \mathrm{i}+(4.0 \mathrm{~m}) \hat{j}\) What are (a) the torque on the particle about the origin, in unit-vector notation, and (b) the angle between the directions of \(\vec{r}\) and \(\vec{F} ?\)

At one instant, force \(\vec{F}=4.0 \mathrm{j} \mathrm{N}\) acts on a \(0.25 \mathrm{~kg}\) object that has position vector \(\vec{r}=(2.0 \mathrm{i}-2.0 \mathrm{k}) \mathrm{m}\) and velocity vector \(\vec{v}=(-5.0 \hat{i}+5.0 \mathrm{k}) \mathrm{m} / \mathrm{s}\). About the origin and in unit-vector notation, what are (a) the object's angular momentum and (b) the torque acting on the object?

A \(2.0 \mathrm{~kg}\) particle-like object moves in a plane with velocity components \(v_{x}=30 \mathrm{~m} / \mathrm{s}\) and \(v_{y}=60 \mathrm{~m} / \mathrm{s}\) as it passes through the point with \((x, y)\) coordinates of \((3.0,-4.0) \mathrm{m} .\) Just then, in unitvector notation, what is its angular momentum relative to (a) the origin and (b) the point located at (-2.0,-2.0) \(\mathrm{m} ?\)

At the instant the displacement of a \(2.00 \mathrm{~kg}\) object relative to the origin is \(\vec{d}=(2.00 \mathrm{~m}) \hat{i}+(4.00 \mathrm{~m}) \mathrm{j}-(3.00 \mathrm{~m}) \mathrm{k},\) its velocity is \(\vec{v}=-(6.00 \mathrm{~m} / \mathrm{s}) \mathrm{i}+(3.00 \mathrm{~m} / \mathrm{s}) \mathrm{j}+(3.00 \mathrm{~m} / \mathrm{s}) \mathrm{k}\) and it is subject to a force \(\vec{F}=(6.00 \mathrm{~N}) \hat{\mathrm{i}}-(8.00 \mathrm{~N}) \mathrm{j}+(4.00 \mathrm{~N}) \mathrm{k} .\) Find (a) the acceleration of the object, (b) the angular momentum of the object about the origin, (c) the torque about the origin acting on the object, and (d) the angle between the velocity of the object and the force acting on the object.

A particle is acted on by two torques about the origin: \(\vec{\tau}_{1}\) has a magnitude of \(2.0 \mathrm{~N} \cdot \mathrm{m}\) and is directed in the positive direction of the \(x\) axis, and \(\vec{\tau}_{2}\) has a magnitude of \(4.0 \mathrm{~N} \cdot \mathrm{m}\) and is directed in the negative direçtion of the \(y\) axis. In unit-vector notation, find \(d \ell / d t,\) where \(\vec{\ell}\) is the angular momentum of the particle about the origin.

At time \(t=0,\) a \(3.0 \mathrm{~kg}\) particle with velocity \(\vec{v}=(5.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}-(6.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}\) is at \(x=3.0 \mathrm{~m}, y=8.0 \mathrm{~m} .\) It is pulled by a \(7.0 \mathrm{~N}\) force in the negative \(x\) direction. About the origin, what are (a) the particle's angular momentum, (b) the torque acting on the particle, and (c) the rate at which the angular momentum is changing?

Three Particles of mass \(m=23 \mathrm{~g}\) are fastened to three rods of length \(d=12 \mathrm{~cm}\) and negligible mass. The rigid assembly rotates around point \(O\) at the angular speed \(\omega=0.85 \mathrm{rad} / \mathrm{s} .\) About \(O\) what are (a) the rotational inertia of the assembly, (b) the magnitude of the angular momentum of the middle particle, and (c) the mag. nitude of the angular momentum of the asssembly?

A sanding disk with rotational inertia \(1.2 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\) is attached to an electric drill whose motor delivers a torque of magnitude \(16 \mathrm{~N} \cdot \mathrm{m}\) about the central axis of the disk. About that axis and with the torque applied for \(33 \mathrm{~ms}\), what is the magnitude of the (a) angular momentum and (b) angular velocity of the disk?

The angular momentum of a flywheel having a rotational inertia of \(0.140 \mathrm{~kg} \cdot \mathrm{m}^{2}\) about its central axis decreases from 3.00 to \(0.800 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}\) in \(1.50 \mathrm{~s}\). (a) What is the magnitude of the average torque acting on the flywheel about its central axis during this period? (b) Assuming a constant angular acceleration, through what angle does the flywheel turn? (c) How much work is done on the wheel? (d) What is the average power of the flywheel?

A disk with a rotational inertia of \(7.00 \mathrm{~kg} \cdot \mathrm{m}^{2}\) rotates like a merry-go-round while undergoing a time-dependent torque given by \(\tau=(5.00+2.00 t) \mathrm{N} \cdot \mathrm{m} .\) At time \(t=1.00 \mathrm{~s},\) its angular momentum is \(5.00 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}\). What is its angular momentum at \(t=3.00 \mathrm{~s} ?\)

Show a rigid structure consisting of a circular hoop of radius \(R\) and mass \(m,\) and a square made of four thin bars, each of length \(R\) and mass \(m\). The rigid structure rotates at a constant speed about a vertical axis, with a period of rotation of 2.5 s. Assuming \(R=0.50 \mathrm{~m}\) and \(m=2.0 \mathrm{~kg},\) calculate (a) the structure's rotational inertia about the axis of rotation and (b) its angular momentum about that axis.

Two skaters, each of mass \(50 \mathrm{~kg}\), approach each other along parallel paths separated by \(3.0 \mathrm{~m}\). They have opposite velocities of \(1.4 \mathrm{~m} / \mathrm{s}\) each. One skater carries one end of a long pole of negligible mass, and the other skater grabs the other end as she passes. The skaters then rotate around the center of the pole. Assume that the friction between skates and ice is negligible. What are (a) the radius of the circle, (b) the angular speed of the skaters, and (c) the kinetic energy of the two-skater system? Next, the skaters pull along the pole until they are separated by \(1.0 \mathrm{~m}\). What then are (d) their angular speed and (e) the kinetic energy of the system? (f) What provided the energy for the increased kinetic energy?

A Texas cockroach of mass \(0.17 \mathrm{~kg}\) runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has radius \(15 \mathrm{~cm}\), rotational inertia \(5.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2},\) and frictionless bearings. The cockroach's speed (relative to the ground) is \(2.0 \mathrm{~m} / \mathrm{s},\) and the lazy Susan turns clockwise with angular speed \(\omega_{0}=2.8 \mathrm{rad} / \mathrm{s} .\) The cockroach finds a bread crumb on the rim and, of course, stops. (a) What is the angular speed of the lazy Susan after the cockroach stops? (b) Is mechanical energy conserved as it stops?

A man stands on a platform that is rotating (without friction) with an angular speed of 1.2 rev/s; his arms are outstretched and he holds a brick in each hand. The rotational thertia of the system consisting of the man, bricks, and platform about the central vertical axis of the platform is \(6.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\). If by moving the bricks the man decreases the rotational inertia of the system to \(2.0 \mathrm{~kg} \cdot \mathrm{m}^{2},\) what are (a) the resulting angular speed of the platform and (b) the ratio of the new kinetic energy of the system to the original kinetic energy? (c) What source provided the added kinetic energy?

The rotational inertia of a collapsing spinning star drops to \(\frac{1}{3}\) its initial value. What is the ratio of the new rotational kinetic energy to the initial rotational kinetic energy?

A track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis (Fig. \(11-48\) ). A toy train of mass \(m\) is placed on the track and, with the system initially at rest,the train's electrical power is turned on. The train reaches speed \(0.15 \mathrm{~m} / \mathrm{s}\) with respect to the track. What is the wheel's angular speed if its mass is \(1.1 \mathrm{~m}\) and its radius is \(0.43 \mathrm{~m} ?\) (Treat it as a hoop, and neglect the mass of the spokes and hub.)

Two disks are mounted (like a merry-go-round) on lowfriction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with rotational inertia \(3.30 \mathrm{~kg} \cdot \mathrm{m}^{2}\) about its central axis, is set spinning counterclockwise at 450 revimin. The second disk, with rotational inertia \(6.60 \mathrm{~kg} \cdot \mathrm{m}^{2}\) about its central axis, is set spinning counterclockwise at 900 revimin. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 900 revimin, what are their (b) angular speed and (c) direction of rotation after they couple together?

The rotor of an electric motor has rotational inertia \(I_{m}=\) \(2.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\) about its central axis. The motor is used to change the orientation of the space probe in which it is mounted. The motor axis is mounted along the central axis of the probe; the probe has rotational inertia \(I_{p}=12 \mathrm{~kg} \cdot \mathrm{m}^{2}\) about this axis. Calculate the number of revolutions of the rotor required to turn the probe through \(30^{\circ}\) about its central axis.

A wheel is rotating freely at angular speed 800 rev/min on a shaft whose rotational inertia is negligible. A second wheel, initially at rest and with twice the rotational inertia of the first, is suddenly coupled to the same shaft. (a) What is the anguLar speed of the resultant combination of the shaft and two wheels? (b) What fraction of the original rotational kinetic energy is lost?

A cockroach of mass \(m\) lies on the rim of a uniform disk of mass \(4.00 \mathrm{~m}\) that can rotate freely about its center like a merrygo-round. Initially the cockroach and disk rotate together with an angular velocity of \(0.260 \mathrm{rad} / \mathrm{s}\). Then the cockroach walks halfway to the center of the disk. (a) What then is the angular velocity of the cockroach-disk system? (b) What is the ratio \(K / K_{0}\) of the new kinetic energy of the system to its initial kinetic energy? (c) What accounts for the change in the kinetic energy?

Shows an overhead view of a ring that can rotate about its center like a merrygo-round. Its outer radius \(R_{2}\) is \(0.800 \mathrm{~m},\) its inner radius \(R_{1}\) is \(R_{2} / 2.00\) its mass \(M\) is \(8.00 \mathrm{~kg}\), and the mass of the crossbars at its center is negligible. It initially rotates at an angular speed of \(8.00 \mathrm{rad} / \mathrm{s}\) with a cat of mass \(m=M / 4.00\) on its outer edge, at radius \(R_{2}\). By how much does the cat increase the kinetic energy of the cat-ring system if the cat crawls to the inner edge, at radius \(R_{1} ?\)

A horizontal vinyl record of mass \(0.10 \mathrm{~kg}\) and radius \(0.10 \mathrm{~m}\) rotates freely about a vertical axis through its center with an angular speed of \(4.7 \mathrm{rad} / \mathrm{s}\) and a rotational inertia of \(5.0 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}\). Putty of mass \(0.020 \mathrm{~kg}\) drops vertically onto the record from above and sticks to the edge of the record. What is the angular speed of the record immediately afterwards?

In a long jump, an athlete leaves the ground with an initial angular momentum that tends to rotate her body forward, threatening to ruin her landing. To counter this tendency, she rotates her outstretched arms to "take up" the angular momentum (Fig. \(11-18\) ). In \(0.700 \mathrm{~s},\) one arm sweeps through 0.500 rev and the other arm sweeps through 1.000 rev. Treat each arm as a thin rod of mass \(4.0 \mathrm{~kg}\) and length \(0.60 \mathrm{~m}\), rotating around one end. In the athlete's reference frame, what is the magnitude of the total angular momentum of the arms around the common rotation axis through the shoulders?

A uniform disk of mass \(10 m\) and radius \(3.0 r\) can rotate freely about its fixed center like a merry-go-round. A smaller uniform disk of mass \(m\) and radius \(r\) lies on top of the larger disk, concentric with it. Initially the two disks rotate together with an angular velocity of \(20 \mathrm{rad} / \mathrm{s}\). Then a slight disturbance causes the smaller disk to slide outward across the larger disk, until the outer edge of the smaller disk catches on the outer edge of the larger disk. Afterward, the two disks again rotate together (without further sliding). (a) What then is their angular velocity about the center of the larger disk? (b) What is the ratio \(K / K_{0}\) of the new kinetic energy of the twodisk system to the system's initial kinetic energy?

A horizontal platform in the shape of a circular disk rotates on a frictionless bearing about a vertical axle through the center of the disk. The platform has a mass of \(150 \mathrm{~kg}\), a radius of \(2.0 \mathrm{~m}\), and a rotational inertia of \(300 \mathrm{~kg} \cdot \mathrm{m}^{2}\) about the axis of rotation. \(\mathrm{A} 60 \mathrm{~kg}\) student walks slowly from the rim of the platform toward the center. If the angular speed of the system is \(1.5 \mathrm{rad} / \mathrm{s}\) when the student starts at the rim, what is the angular speed when she is \(0.50 \mathrm{~m}\) from the center?

An overhead view of a thin uniform rod of length \(0.800 \mathrm{~m}\) and mass \(M\) rotating horizontally at angular speed \(20.0 \mathrm{rad} / \mathrm{s}\) about an axis through its center. A particle of mass \(M / 3,00\) initially attached to one end is ejected from the rod and travels along a path that is perpendicular to the rod at the instant of ejection. If the particle's speed \(v_{p}\) is \(6.00 \mathrm{~m} / \mathrm{s}\) greater than the speed of the rod end just after ejection, what is the value of \(v_{p} ?\)

A \(1.0 \mathrm{~g}\) bullet is fired into a \(0.50 \mathrm{~kg}\) block attached to the end of a \(0.60 \mathrm{~m}\) nonuniform rod of mass \(0.50 \mathrm{~kg} .\) The block-rod-bullet system then rotates in the plane of the figure, about a fixed axis at \(A .\) The rotational inertia of the rod alone about that axis at \(A\) is \(0.060 \mathrm{~kg} \cdot \mathrm{m}^{2}\). Treat the block as a particle. (a) What then is the rotational inertia of the block-rod-bullet system about point \(A ?\) (b) If the angular speed of the system about \(A\) just after impact is \(4.5 \mathrm{rad} / \mathrm{s},\) what is the bullet's speed just before impact?

A ballerina begins a tour jeté (Fig. \(11-19 a\) ) with angular speed \(\omega_{i}\) and a rotational inertia consisting of two parts:\(I_{\mathrm{leg}}=1.44 \mathrm{~kg} \cdot \mathrm{m}^{2}\) for her leg extended outward at angle \(\theta=90.0^{\circ}\) to her body and \(I_{\text {trunk }}=0.660 \mathrm{~kg} \cdot \mathrm{m}^{2}\) for the rest of her body (primarily her trunk ). Near her maximum height she holds both legs at angle \(\theta=30.0^{\circ}\) to her body and has angular speed \(\omega_{f}(\) Fig. \(11-19 b)\). Assuming that \(I_{\text {trunk }}\) has not changed, what is the ratio \(\omega_{f} / \omega_{i} ?\)

Two \(2.00 \mathrm{~kg}\) balls are attached to the ends of a thin rod of length \(50.0 \mathrm{~cm}\) and negligible mass. The rod is free to rotate in a vertical plane without friction about a horizontal axis through its center. With the rod initially horizontal (Fig.\(11-57),\) a \(50.0 \mathrm{~g}\) wad of wet putty drops onto one of the balls, hitting it with a speed of \(3.00 \mathrm{~m} / \mathrm{s}\) and then sticking to it. (a) What is the angular speed of the system just after the putty wad hits? (b) What is the ratio of the kinetic energy of the system after the collision to that of the putty wad just before? (c) Through what angle will the system rotate before it momentarily stops?

A small \(50 \mathrm{~g}\) block slides down a frictionless surface through height \(h=20 \mathrm{~cm}\) and then sticks to a uniform rod of mass \(100 \mathrm{~g}\) and length \(40 \mathrm{~cm} .\) The rod pivots about point \(O\) through angle \(\theta\) before momentarily stopping. Find \(\theta\).

Is an over head view of a thin uniform rod of length \(0.600 \mathrm{~m}\) and mass \(M\) rotating horizontally at \(80.0 \mathrm{rad} / \mathrm{s}\) counterclockwise about an axis through its center. A particle of mass \(M / 3.00\) and traveling horizontally at speed \(40.0 \mathrm{~m} / \mathrm{s}\) hits the rod and sticks. The particle's path is perpendicular to the rod at the instant of the hit, at a distance \(d\) from the rod's center. (a) At what value of \(d\) are rod and particle stationary after the hit? (b) In which direction do rod and particle rotate if \(d\) is greater than this value?

A top spins at 30 rev/s about an axis that makes an angle of \(30^{\circ}\) with the vertical. The mass of the top is \(0.50 \mathrm{~kg}\), its rotational inertia about its central axis is \(5.0 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2},\) and its center of mass is \(4.0 \mathrm{~cm}\) from the pivot point. If the spin is clockwise from an overhead view, what are the (a) precession rate and (b) direction of the precession as viewed from overhead?