A man opens a 1.00-m wide door by pushing on it with a force of 50.0 N directed perpendicular to its surface. What magnitude of torque does he apply about an axis through the hinges if the force is applied (a) at the center of the door? (b) at the edge farthest from the hinges?

Salamat A.

Numerade Educator

A worker applies a torque to a nut with a wrench 0.500 m long. Because of the cramped space, she must exert a force upward at an angle of 60.0° with respect to a line from the nut through the end of the wrench. If the force she exerts has magnitude 80.0 N, what magnitude torque does she apply to the nut?

Averell H.

Carnegie Mellon University

The fishing pole in Figure P8.3 makes an angle of 20.0° with the horizontal. What is the magnitude of the torque exerted by the fish about an axis perpendicular to the page and passing through the angler’s hand if the fish pulls on the fishing line with a force $\overrightarrow{\mathbf{F}}=1.00 \times 10^{2} \mathrm{N}$ at an angle $37.0^{\circ}$ below the horizontal? The force is applied at a point 2.00 $\mathrm{m}$ from the angler's bands.

Salamat A.

Numerade Educator

Find the net torque on the wheel in Figure P8.4 about the axle through $O$ perpendicular to the page, taking $a=10.0 \mathrm{cm}$ and $b=25.0 \mathrm{cm} .$

Averell H.

Carnegie Mellon University

Calculate the net torque (magnitude and direction) on the beam in Figure P 8.5 about (a) an axis through $O$ perpendicular to the page and (b) an axis through $C$ perpendicular to the page.

Salamat A.

Numerade Educator

A dental bracket exerts a horizontal force of 80.0 N on a tooth at point $B$ in Figure P 8.6. What is

the torque on the root of the tooth about point $A?$

Averell H.

Carnegie Mellon University

A simple pendulum consists of a small object of mass 3.0 kg hanging at the end of a 2.0-m-long

light string that is connected to a pivot point. (a) Calculate the magnitude of the torque (due to the force of gravity) about this pivot point when the string makes a 5.0° angle with the vertical. (b) Does the torque increase or decrease as the angle increases? Explain.

Salamat A.

Numerade Educator

Consider the following mass distribution, where $x-$ and $y-$coordinates are given in meters: 5.0 kg at (0.0, 0.0) m, 3.0 kg at (0.0, 4.0) m, and 4.0 kg at (3.0, 0.0) m. Where should a fourth object of 8.0 kg be placed so that the center of mass of the four-object arrangement will be at (0.0, 0.0) m?

Averell H.

Carnegie Mellon University

Two bowling balls are at rest on top of a uniform wooden plank with their centers of mass located as in Figure P 8.9. The plank has a mass of 5.00 kg and is 1.00 m long. Find the horizontal distance from the left end of the plank to the center of mass of the plank–bowling balls system.

Salamat A.

Numerade Educator

Three solid, uniform boxes are aligned as in Figure P 8.10. Find the $x$ - and $y$ -coordinates of the center of mass of the three boxes, measured from the bottom left corner of box A.

Averell H.

Carnegie Mellon University

Find the $x$ - and $y$ -coordinates of the center of gravity of a 4.00 -ft by 8.00 -ft uniform sheet of plywood with the upper right quadrant removed as shown in Figure P 8.11. Hint: The mass of any segment of the plywood sheet is proportional to the area of that segment.

Salamat A.

Numerade Educator

Find the $x$- and $y$-coordinates of the center of gravity for the boomerang in Figure P 8.12a, modeling the boomerang as in Figure P 8.12b, where each uniform leg of the model has a length of 0.300 m and a mass of 0.250 kg. (Note: Treat the legs like thin rods.)

Averell H.

Carnegie Mellon University

A block of mass $m=1.50 \mathrm{kg}$ is at rest on a ramp of mass $M=$ 4.50 kg which, in turn, is at rest on a frictionless horizontal surface (Fig. P8.13a). The block and the ramp are aligned so that each has its center of mass located at $x=0 .$ When released, the block slides down the ramp to the left and the ramp, also free to slide on the frictionless surface, slides to the right as in Figure P 8.13 b. Calculate $x_{\text { ramp }}$ the distance the ramp has moved to the right, when $x_{\text { block }}=-0.300 \mathrm{m} .$

Salamat A.

Numerade Educator

The Xanthar mother ship locks onto an enemy cruiser with its tractor beam (Fig. P 8.14); each ship is at rest in deep space with no propulsion following a devastating battle. The mother ship is at $x=0$ when its tractor beams are first engaged, a distance $d=215$ xiles from the cruiser. Determine the $x$ -position in xiles of the two spacecraft when the tractor beam has pulled them together. Model each spacecraft as a point particle with the mothership of mass $M=185$ xons and the cruiser of mass $m=20.0$ xons.

Averell H.

Carnegie Mellon University

A hiker inspects a tree frog sitting on a small stick in his hand. Suddenly startled, the hiker drops the stick from rest at a height of 1.85 m above the ground and, at the same instant, the frog leaps vertically upward, pushing the stick down so that it hits the ground 0.450 s later. Find the height of the frog at the instant the stick hits the ground if the frog and the stick have masses of 7.25 g and 4.50 g, respectively. (Hint: Find the center-of-mass height at $t=0.450$ s for the frog-stick system and then use the definition of center of mass to solve for the frog's height.)

Ankit P.

Numerade Educator

Spectators watch a bicycle stunt rider travel off the end of a 60.0° ramp, rise to the top of his trajectory and, at that instant, suddenly push his bike away from him so that he falls vertically straight down, reaching the ground 0.550 s later. How far from the rider does the bicycle land if the rider has mass $M=72.0 \mathrm{kg}$ and the bike has mass $m=12.0 \mathrm{kg} ?$ Neglect air resistance and assume the ground is level.

Averell H.

Carnegie Mellon University

The arm in Figure P8.17 weighs 41.5 N. The force of gravity acting on the arm acts through point A. Determine the magnitudes of the tension force $\overrightarrow{\mathbf{F}}_{t}$ in the deltoid muscle and the force $\overrightarrow{\mathbf{F}}_{s}$ exerted by the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown.

Salamat A.

Numerade Educator

A uniform 35.0-kg beam of length $\ell=5.00 \mathrm{m}$ is supported by a vertical rope located $d=1.20 \mathrm{m}$ from cal rope located d 5 1.20 m from its left end as in Figure P 8.18. The

right end of the beam is supported by a vertical column. Find (a) the tension in the rope and (b) the force that the column exerts on the right end of the beam.

Averell H.

Carnegie Mellon University

A cook holds a 2.00-kg carton of milk at arm’s length (Fig. P8.19). What force $\overrightarrow{\mathbf{F}}_{B}$ must be exerted by the biceps muscle? (Ignore the weight of the forearm.)

Salamat A.

Numerade Educator

A meter stick is found to balance at the 49.7-cm mark when placed on a fulcrum. When a 50.0-gram mass is attached at the 10.0-cm mark, the fulcrum must be moved to the 39.2-cm mark for balance. What is the mass of the meter stick?

Averell H.

Carnegie Mellon University

In exercise physiology studies, it is sometimes important to determine the location of a person’s center of gravity. This can be done with the arrangement shown in Figure P 8.21. A light plank rests on two scales that read $F_{g 1}=380 .$ N and $F_{g 2}=320 .$ N. The scales are separated by a distance of 2.00 m. How far from the woman's feet is her center of gravity?

Salamat A.

Numerade Educator

A beam resting on two pivots has a length of $L=6.00 \mathrm{m}$ and mass $M=90.0 \mathrm{kg}$ . The pivot under the left end exerts a normal force $n_{1}$ on the beam, and the second pivot placed a distance $\ell=4.00 \mathrm{m}$ from the left end exerts a normal force $n_{2}$ A woman of mass $m=55.0 \mathrm{kg}$ steps onto the left end of the beam and begins walking to the right as in Figure P 8.22. The goal is to find the woman’s position when the beam begins to tip. (a) Sketch a free-body diagram, labeling the gravitational and normal forces acting on the beam and placing the woman $x$ meters to the right of the first pivot, which is the origin. (b) Where is the woman when the normal force $n_{1}$ is the greatest? (c) What is $n_{1}$ when the beam is about to tip? (d) Use the force equation of equilibrium to find the value of $n_{2}$ when the beam is about to tip. (e) Using the result of part (c) and the torque equilibrium equation, with torques computed around the second pivot point, find the woman’s position when the beam is about to tip. (f) Check the answer to part (e) by computing torques around the first pivot point. Except for possible slight differences due to rounding, is the answer the same?

Averell H.

Carnegie Mellon University

A person bending forward to lift a load “with his back” (Fig. P 8.23a) rather than “with his knees” can be injured by large forces exerted on the muscles and vertebrae. The spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, and to understand why back problems are common among humans, consider the model shown in Figure P 8.23b of a person bending forward to lift a 200.-N object. The spine and upper body are represented as a uniform horizontal rod of weight 350. N, pivoted at the base of the spine. The erector spinalis muscle, attached at a point two-thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is 12.0°. Find (a) the tension in the back muscle and (b) the compressional force in the spine.

Salamat A.

Numerade Educator

When a person stands on tiptoe (a strenuous position), the position of the foot is as shown in Figure P8.24a. The total gravitational force on the body, $\overrightarrow{\mathbf{F}}_{g},$ is supported by the force $\overrightarrow{\mathbf{n}}$ exerted by the floor on the toes of one foot. A mechanical

model of the situation is shown in Figure $P 8.24 \mathrm{b},$ where $\overrightarrow{\mathbf{T}}$ is the force exerted by the Achilles tendon on the foot and $\overrightarrow{\mathbf{R}}$ is the force exerted by the tibia on the foot. Find the values of $T$ $R,$ and $\theta$ when $F_{g}=n=700 . \mathrm{N}$ .

Sheh Lit C.

University of Washington

A 500.-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform, 100.-N rod as indicated in Figure P8.25. The left end of the rod is supported by a hinge, and the right end is supported by a thin cable making a 30.0° angle with the vertical. (a) Find the tension $T$ in the cable. (b) Find the horizontal and vertical components of force exerted on the left end of the rod by the hinge.

Salamat A.

Numerade Educator

A window washer is standing on a scaffold supported by a vertical rope at each end. The scaffold weighs 200 N and is 3.00 m long. What is the tension in each rope when the 700-N worker stands 1.00 m from one end?

Averell H.

Carnegie Mellon University

A uniform plank of length 2.00 m and mass 30.0 kg is supported by three ropes, as indicated by the blue vectors in Figure P 8.27. Find the tension in each rope when a 700.-N person is $d=0.500 \mathrm{m}$ from the left end.

Salamat A.

Numerade Educator

A hungry bear weighing 700. N walks out on a beam in an attempt to retrieve a basket of goodies

hanging at the end of the beam (Fig. P8.28). The beam is uniform, weighs 200. N, and is 6.00 m long, and it is supported by a wire at an angle of u 5 60.0°. The basket weighs 80.0 N. (a) Draw a force diagram for the beam. (b) When the bear is at $x=1.00 \mathrm{m},$ find the tension in the wire supporting the beam and the components of the force exerted by the wall on the left end of the beam. (c) If the wire can withstand a maximum tension of 900. N, what is the maximum distance the bear can walk before the wire breaks?

Averell H.

Carnegie Mellon University

Figure P 8.29 shows a uniform beam of mass $m$ pivoted at its lower end, with a horizontal spring attached between its top end and a vertical wall. The beam makes an angle $\theta$ with the

horizontal. Find expressions for (a) the distance $d$ the spring is stretched from equilibrium and

(b) the components of the force exerted by the pivot on the beam.

Salamat A.

Numerade Educator

A strut of length $L=3.00 \mathrm{m}$ and mass $m=$ 16.0 kg is held by a cable at an angle of u 5 30.0° with respect to the horizontal as shown in Figure P 8.30. (a) Sketch a force diagram, indicating all the forces and their placement on the strut. (b) Why is the hinge a good place to use for calculating torques? (c) Write the condition for rotational equilibrium symbolically, calculating the torques around the hinge. (d) Use the torque equation to calculate the tension in the cable. (e) Write the $x$ - and $y$-components of Newton's second law for equilibrium. (f) Use the force equation to find the $x$ - and $y$ -components of the force on the hinge. (g) Assuming the strut position is to remain the same, would it be advantageous to attach the cable higher up on the wall? Explain the benefit in terms of the force on the hinge and cable tension.

Averell H.

Carnegie Mellon University

A refrigerator of width $w$ and height $h$ rests on a rough incline as in Figure $P 8.31$ Find an expression for the maximum value $\theta$ can have before the refrigerator tips over. Note, the contact point between the refrigerator and incline shifts as $\theta$ increases and treat the refrigerator as a uniform box.

Salamat A.

Numerade Educator

Write the necessary equations of equilibrium of the object shown in Figure P8.32. Take the origin of the torque equation about an axis perpendicular to the page through the point $O.$

Averell H.

Carnegie Mellon University

The chewing muscle, the masseter, is one of the strongest in the human body. It is attached to the mandible (lower jawbone) as shown in Figure P8.33a. The jawbone is pivoted about a socket just in front of the auditory canal. The forces acting on the jawbone are equivalent to those acting on the curved bar in Figure $P 8.33 \mathrm{b} . \overrightarrow{\mathbf{F}}_{C}$ is the force exerted by the

food being chewed against the jawbone, $\overrightarrow{\mathbf{T}}$ is the force of tension in the masseter, and $\overrightarrow{\mathbf{R}}$ is the force exerted by the socket on the mandible. Find $\overrightarrow{\mathbf{T}}$ and $\overrightarrow{\mathbf{R}}$ for a person who bites down on a piece of steak with a force of 50.0 N.

Salamat A.

Numerade Educator

A 1200-N uniform boom at $\phi=65^{\circ}$ to the horizontal is sup- ported by a cable at an angle $\theta=$ $25.0^{\circ}$ to the horizontal as shown in Figure $P 8.34$ . The boom is pivoted at the bottom, and an object of weight w = 2000 N hangs from its top. Find (a) the tension in the support cable and (b) the components of the reaction force exerted by the pivot on the boom.

Averell H.

Carnegie Mellon University

The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (Fig. P8.35a). The forces on the lower leg when the leg is extended are

modeled as in Figure P8.35b, where $\overrightarrow{\mathbf{T}}$ is the force of tension in the tendon, $\vec{w}$ is the force of gravity acting on the lower leg, and $\overrightarrow{\mathbf{F}}$ is the force of gravity acting on the foot. Find $\overrightarrow{\mathbf{T}}$ when the tendon is at an angle of 25.0° with the tibia, assuming that $w=30.0 \mathrm{N}, F=12.5 \mathrm{N},$ and the leg is extended at an angle $\theta$ of $40.0^{\circ}$ with the vertical. Assume that the center of gravity of the lower leg is at its center and that the tendon attaches to the lower leg at a point one-fifth of the way down the leg.

Salamat A.

Numerade Educator

One end of a uniform 4.0-m-long rod of weight $w$ is supported by a cable at an angle of u 5 37° with

the rod. The other end rests against a wall, where it is held by friction. (See Fig. P8.36.) The coefficient of static friction between the wall and the rod is $\mu_{s}=0.50$ Determine the minimum

distance $x$ from point A at which an additional weight $w$ (the same as the weight of the rod) can be hung without causing the rod to slip at point A.

Averell H.

Carnegie Mellon University

Four objects are held in position at the corners of a rectangle by light rods as shown in Figure P8.37. Find the moment of inertia of the system about (a) the $x$ -axis, $(\mathrm{b})$ the $y$ -axis, and $(\mathrm{c})$ an axis through $O$ and perpendicular to the page.

Salamat A.

Numerade Educator

If the system shown in Figure P8.37 is set in rotation about each of the axes mentioned in Problem 37, find the torque that will produce an angular acceleration of $1.50 \mathrm{rad} / \mathrm{s}^{2}$ in each case.

Averell H.

Carnegie Mellon University

A large grinding wheel in the shape of a solid cylinder of radius 0.330 m is free to rotate on a frictionless, vertical axle. A constant tangential force of 250. N applied to its edge causes the wheel to have an angular acceleration of 0.940 $\operatorname{rad} / \mathrm{s}^{2}.$ (a) What is the moment of inertia of the wheel? (b) What is the mass of the wheel? (c) If the wheel starts from rest, what is its angular velocity after 5.00 s have elapsed, assuming the force is acting during that time?

Salamat A.

Numerade Educator

An oversized yo-yo is made from two identical solid disks each of mass $M=2.00 \mathrm{kg}$ and radius $R=10.0 \mathrm{cm} .$ The two disks are joined by a solid cylinder of radius $r=4.00 \mathrm{cm}$ and mass $m=1.00 \mathrm{kg}$ as in Figure $\mathrm{P} 8.40 .$ Take the center of the cylinder as the axis of the system, with positive torques directed to the left along this axis. All torques and angular variables are to be calculated around this axis. Light string is wrapped around the cylinder, and the system is then allowed to drop from rest. (a) What is the moment of inertia of the system? Give a symbolic answer. (b) What torque does gravity exert on the system with respect to the given axis? (c) Take downward as the negative coordinate direction. As depicted in Figure P8.40, is the torque exerted by the tension positive or negative? Is the angular acceleration positive or negative? What about the translational acceleration? (d) Write an equation for the angular acceleration a in terms of the translational acceleration a and radius r. (Watch the sign!) (e) Write Newton’s second law for the system in terms of $m, M, a, T,$ and $g .(\text { f) Write Newton's }$ second law for rotation in terms of $I, \alpha, T,$ and $r$ . $r$ (g) Eliminate $\alpha$ from the rotational second law with the expression found in part (d) and find a symbolic expression for the acceleration $a$ in terms of $m, M, g, r,$ and $R$ (h) What is the numeric value for the system's acceleration? (i) What is the tension in the string? (j) How long does it take the system to drop 1.00 $\mathrm{m}$ from rest?

Dading C.

Numerade Educator

An approximate model for a ceiling fan consists of a cylindrical disk with four thin rods extending from the disk’s center, as in Figure P8.41. The disk has mass 2.50 kg and radius 0.200 m. Each rod has mass 0.850 kg and is 0.750 m long. (a) Find the ceiling fan’s moment of inertia about a vertical axis through the disk’s center. (b) Friction exerts a constant torque of magnitude 0.115 N ? m on the fan as it rotates. Find the magnitude of the constant torque provided by the fan’s motor if the fan starts from rest and takes 15.0 s and 18.5 full revolutions to reach its maximum speed.

Salamat A.

Numerade Educator

A potter’s wheel having a radius of 0.50 m and a moment of inertia of 12 $\mathrm{kg} \cdot \mathrm{m}^{2}$ is rotating freely at 50 rev/min. The potter can stop the wheel in 6.0 s by pressing a wet rag against the rim and exerting a radially inward force of 70 N. Find the effective coefficient of kinetic friction between the wheel and the wet rag.

Averell H.

Carnegie Mellon University

A model airplane with mass 0.750 kg is tethered by a wire so that it flies in a circle 30.0 m in radius. The airplane engine provides a net thrust of 0.800 N perpendicular to the tethering wire. (a) Find the torque the net thrust produces about the center of the circle. (b) Find the angular acceleration of the airplane when it is in level flight. (c) Find the linear acceleration of the airplane tangent to its flight path.

Salamat A.

Numerade Educator

A bicycle wheel has a diameter of 64.0 cm and a mass of 1.80 kg. Assume that the wheel is a hoop with all the mass concentrated on the outside radius. The bicycle is placed on a stationary stand, and a resistive force of 120 N is applied tangent to the rim of the tire. (a) What force must be applied by a chain passing over a 9.00-cm-diameter sprocket to give the wheel an acceleration of 44.50 $\mathrm{rad} / \mathrm{s}^{2} ?$ (b) What force is required if you shift to a 5.60- cm - diameter sprocket?

Averell H.

Carnegie Mellon University

A 150.-kg merry - go - round in the shape of a uniform, solid, horizontal disk of radius 1.50 m is set in motion by wrapping a rope about the rim of the disk and pulling on the rope. What constant force must be exerted on the rope to bring the merry - go - round from rest to an angular speed of 0.500 rev/s in 2.00 s?

Salamat A.

Numerade Educator

An Atwood’s machine consists of blocks of masses $m_{1}=10.0 \mathrm{kg}$ and $m_{2}=20.0 \mathrm{kg}$ attached by a cord and $m_{2}=20.0 \mathrm{kg}$ attached by a cord running over a pulley as in Figure P8.46. The pulley is a solid cylinder with mass $M=8.00 \mathrm{kg}$ and radius $r=0.200 \mathrm{m} .$ The block of mass $m_{2}$ is allowed to drop, and the cord turns the pulley without slipping. (a) Why must the tension $T_{2}$ be greater than the tension $T_{1} ?$ (b) What is the acceleration of the system, assuming the pulley axis is frictionless? (c) Find the tensions $T_{1}$ and $T_{2}$ .

Averell H.

Carnegie Mellon University

The uniform thin rod in Figure P8.47 has mass $M=3.50 \mathrm{kg}$ and length $L=1.00 \mathrm{m}$ and is free to rotate on a frictionless pin. At the instant the rod is released from rest in the horizontal position, find the magnitude of (a) the rod’s angular acceleration, (b) the tangential acceleration of the rod’s center of mass, and (c) the tangential acceleration of the rod’s free end.

Salamat A.

Numerade Educator

A 2.50-kg solid, uniform disk rolls without slipping across a level surface, translating at 3.75 m/s. If the disk’s radius is 0.100 m, find its (a) translational kinetic energy and (b) rotational kinetic energy.

Averell H.

Carnegie Mellon University

A horizontal 800.-N merry-go-round of radius 1.50 m is started from rest by a constant horizontal force of 50.0 N applied tangentially to the merry-go-round. Find the kinetic energy of the merry-go-round after 3.00 s. (Assume it is a solid cylinder.)

Salamat A.

Numerade Educator

Four objects—a hoop, a solid cylinder, a solid sphere, and a thin, spherical shell—each have a mass of 4.80 kg and a radius of 0.230 m. (a) Find the moment of inertia for each object as it rotates about the axes shown in Table 8.1. (b) Suppose each object is rolled down a ramp. Rank the translational speed of each object from highest to lowest. (c) Rank the objects’ rotational kinetic energies from highest to lowest as the objects roll down the ramp.

Averell H.

Carnegie Mellon University

A light rod of length $\ell=1.00 \mathrm{m}$ rotates about an axis perpendicular to its length and passing through its center as in Figure P 8.51 . Two particles of masses $m_{1}=4.00 \mathrm{kg}$ and $m_{2}=$ $3.00 \mathrm{kg}$ are connected to the ends of the rod. (a) Neglecting the mass of the rod, what is the system’s kinetic energy when its angular speed is 2.50 rad/s? (b) Repeat the problem, assuming the mass of the rod is taken to be 2.00 kg.

Salamat A.

Numerade Educator

A 240-N sphere 0.20 m in radius rolls without slipping 6.0 m down a ramp that is inclined at 37° with the horizontal. What is the angular speed of the sphere at the bottom of the slope if it starts from rest?

Averell H.

Carnegie Mellon University

A solid, uniform disk of radius 0.250 m and mass 55.0 kg rolls down a ramp of length 4.50 m that makes an angle of 15.0° with the horizontal. The disk starts from rest from the top of the ramp. Find (a) the speed of the disk’s center of mass when it reaches the bottom of the ramp and (b) the angular speed of the disk at the bottom of the ramp.

Salamat A.

Numerade Educator

A car is designed to get its energy from a rotating solid-disk flywheel with a radius of 2.00 $\mathrm{m}$ and a mass of $5.00 \times 10^{2} \mathrm{kg}$ . Before a trip, the flywheel is attached to an electric motor, which brings the flywheel's rotational speed up to $5.00 \times 10^{3} \mathrm{rev} / \mathrm{min}$ . (a) Find the kinetic energy stored in the flywheel. (b) If the fly-wheel is to supply energy to the car as a 10.0 - hp motor would, find the length of time the car could run before the flywheel would have to be brought back up to speed.

Averell H.

Carnegie Mellon University

The top in Figure P8.55 has a moment of inertia of $4.00 \times$ $10^{-4} \mathrm{kg} \cdot \mathrm{m}^{2}$ and is initially at rest. It is free to rotate about a stationary axis $A A^{\prime} \cdot A$ string wrapped around a peg along the axis of the top is pulled in such a manner as to maintain a constant tension of 5.57 N in the string. If the string does not slip while wound around the peg, what is the angular speed of the top after 80.0 cm of string has been pulled off the peg? Hint: Consider the work that is done.

Salamat A.

Numerade Educator

A constant torque of 25.0 $\mathrm{N} \cdot \mathrm{m}$ is applied to a grindstone whose moment of inertia is 0.130 $\mathrm{kg} \cdot \mathrm{m}^{2} .$ Using energy principles and neglecting friction, find the angular speed after the grindstone has made 15.0 revolutions. Hint: The angular equivalent of $W_{\mathrm{net}}=\frac{1}{2} m v_{f}^{2}-\frac{1}{2} m v_{i}^{2}$ is $W_{\mathrm{net}}=\tau \Delta \theta=$ $\frac{1}{2} I \omega_{f}^{2}-\frac{1}{2} I \omega_{i}^{2} .$ You should convince yourself that this last relationship is correct.

Averell H.

Carnegie Mellon University

A 10.0-kg cylinder rolls without slipping on a rough surface. At an instant when its center of gravity has a speed of 10.0 m/s, determine (a) the translational kinetic energy of its center of gravity, (b) the rotational kinetic energy about its center of gravity, and (c) its total kinetic energy.

Salamat A.

Numerade Educator

Use conservation of energy to determine the angular speed of the spool shown in Figure P8.58 after the 3.00-kg bucket has fallen 4.00 m, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds.

Averell H.

Carnegie Mellon University

A 2.00-kg solid, uniform ball of radius 0.100 m is released from rest at point A in Figure P8.59, its center of gravity a distance of 1.50 m above the ground. The ball rolls without slipping to the bottom of an incline and back up to point B where it is launched vertically into the air. The ball rises to its maximum height $h_{\max }$ at point $\mathrm{C}$ At point $\mathrm{B},$ find the ball's (a) translational speed $v_{\mathrm{B}}$ and $(\mathrm{b})$ rotational speed $\omega_{\mathrm{B}}$ . At point $\mathrm{C},$ find the ball's (c) rotational speed $\omega_{\mathrm{C}}$ and (d) maximum height $h_{\max }$ of its center of gravity.

Dading C.

Numerade Educator

Each of the following objects has a radius of 0.180 m and a mass of 2.40 kg, and each rotates about an axis through its center (as in Table 8.1) with an angular speed of 35.0 rad/s. Find the magnitude of the angular momentum of each object. (a) a hoop (b) a solid cylinder (c) a solid sphere (d) a hollow spherical shell

Averell H.

Carnegie Mellon University

A metal hoop lies on a horizontal table, free to rotate about a fixed vertical axis through its center while a constant tangential force applied to its edge exerts a torque of magnitude $1.25 \times 10^{-2} \mathrm{N} \cdot \mathrm{m}$ for 2.00 $\mathrm{s} .$ (a) Calculate the magnitude of the hoop’s change in angular momentum. (b) Find the change in the hoop’s angular speed if its mass and radius are 0.250 kg and 0.100 m, respectively.

Salamat A.

Numerade Educator

A disk of mass $m$ is spinning freely at 6.00 rad/s when a second identical disk, initially not spinning, is dropped onto it so that their axes coincide. In a short time the two disks are corotating. (a) What is the angular speed of the new system? (b) If a third such disk is dropped on the first two, find the final angular speed of the system.

Averell H.

Carnegie Mellon University

(a) Calculate the angular momentum of Earth that arises from its spinning motion on its axis, treating Earth as a uniform solid sphere. (b) Calculate the angular momentum of Earth that arises from its orbital motion about the Sun, treating Earth as a point particle.

Salamat A.

Numerade Educator

A 0.00500-kg bullet traveling horizontally with a speed of $1.00 \times 10^{3} \mathrm{m} / \mathrm{s}$ enters an $18.0 \mathrm{-kg}$ door, embedding itself 10.0 cm from the side opposite the hinges as in Figure P8.64. The 1.00-m-wide door is free to swing on its hinges. (a) Before it hits the door, does the bullet have angular momentum relative the door’s axis of rotation? Explain. (b) Is mechanical energy conserved in this collision? Answer without doing a calculation. (c) At what angular speed does the door swing open immediately after the collision? (The door has the same moment of inertia as a rod with axis at one end.) (d) Calculate the energy of the door–bullet system and determine whether it is less than or equal to the kinetic energy of the bullet before the collision.

Dading C.

Numerade Educator

A light rigid rod of length $\ell=1.00 \mathrm{m}$ rotates about an axis perpendicular to its length and through its center, as shown in Figure $\mathrm{P} 8.51 .$ Two particles of masses $m_{1}=4.00 \mathrm{kg}$ and $m_{2}=3.00 \mathrm{kg}$ are connected to the ends of the rod. What is the angular momentum of the system if the speed of each particle is 5.00 $\mathrm{m} / \mathrm{s}$ ? (Neglect the rod's mass.)

Salamat A.

Numerade Educator

Halley’s comet moves about the Sun in an elliptical orbit, with its closest approach to the Sun being 0.59 AU and its greatest distance being 35 AU (1 AU is the Earth–Sun distance). If the comet’s speed at closest approach is 54 km/s, what is its speed when it is farthest from the Sun? You may neglect any change in the comet’s mass and assume that its angular momentum about the Sun is conserved.

Guilherme B.

Numerade Educator

A student holds a spinning bicycle wheel while sitting motionless on a stool that is free to rotate about a vertical axis through its center (Fig. P8.67). The wheel spins with an angular speed of 17.5 rad/s and its initial angular momentum is directed up. The wheel's moment of inertia is 0.150 $\mathrm{kg} \cdot \mathrm{m}^{2}$ and the moment of inertia for the student plus stool is $3.00 \mathrm{kg} \cdot \mathrm{m}^{2}$ . (a) Find the student's final angular speed after he turns the wheel over so that it spins at the same speed but with its angular momentum directed down. (b) Will the student's final angular momentum be directed up or down?

Salamat A.

Numerade Educator

A 60.0-kg woman stands at the rim of a horizontal turntable having a moment of inertia of 500 $\mathrm{kg} \cdot \mathrm{m}^{2}$ and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to Earth. (a) In what direction and with what angular speed does the turntable rotate? (b) How much work does the woman do to set herself and the turntable into motion?

Averell H.

Carnegie Mellon University

A solid, horizontal cylinder of mass 10.0 kg and radius 1.00 m rotates with an angular speed of 7.00 rad/s about a fixed vertical axis through its center. A 0.250-kg piece of putty is dropped vertically onto the cylinder at a point 0.900 m from the center of rotation and sticks to the cylinder. Determine the final angular speed of the system.

Salamat A.

Numerade Educator

A student sits on a rotating stool holding two 3.0-kg objects. When his arms are extended horizontally, the objects are 1.0 m from the axis of rotation and he rotates with an angular speed of 0.75 rad/s. The moment of inertia of the student plus stool is 3.0 $\mathrm{kg} \cdot \mathrm{m}^{2}$ and is assumed to be constant. The student then pulls in the objects horizontally to 0.30 $\mathrm{m}$ from the rotation axis. ( a) Find the new angular speed of the student. (b) Find the kinetic energy of the student before and after the objects are pulled in.

Averell H.

Carnegie Mellon University

The puck in Figure P 8.71 has a mass of 0.120 kg. Its original distance from the center of rotation is 40.0 cm, and it moves with a speed of 80.0 cm/s. The string is pulled downward 15.0 cm through the hole in the frictionless table. Determine the work done on the puck. Hint: Consider the change in kinetic energy of the puck.

Salamat A.

Numerade Educator

A space station shaped like a giant wheel has a radius of 100 m and a moment of inertia of $5.00 \times 10^{8} \mathrm{kg} \cdot \mathrm{m}^{2} .$ A crew of 150 lives on the rim, and the station is rotating so that the crew experiences an apparent acceleration of 1g (Fig. P8.72). When 100 people move to the center of the station for a union meeting, the angular speed changes. What apparent acceleration is experienced by the managers remaining at the rim? Assume the average mass of a crew member is 65.0 kg.

Averell H.

Carnegie Mellon University

A cylinder with moment of inertia $I_{1}$ rotates with angular velocity $\omega_{0}$ about a frictionless vertical axle. A second cylinder, with moment of inertia $I_{2},$ initially not rotating, drops onto the first cylinder (Fig. P 8.73). Because the surfaces are rough, the two cylinders eventually reach the same angular speed $\omega .$ (a) Calculate $\omega$ . (b) Show that kinetic energy is lost in this situation, and calculate the ratio of the final to the initial kinetic energy.

Salamat A.

Numerade Educator

A particle of mass 0.400 kg is attached to the 100-cm mark of a meter stick of mass 0.100 kg. The meter stick rotates on a horizontal, frictionless table with an angular speed of 4.00 rad/s. Calculate the angular momentum of the system when the stick is pivoted about an axis (a) perpendicular to the table through the 50.0-cm mark and (b) perpendicular to the table through the 0-cm mark.

Averell H.

Carnegie Mellon University

A typical propeller of a turbine used to generate electricity from the wind consists of three blades as in Figure P8.75. Each blade has a length of $L=35 \mathrm{m}$ and a mass of $m=420 \mathrm{kg} .$ The propeller rotates at the rate of 25 rev/min. (a) Convert the angular speed of the propeller to units of rad/s. Find (b) the moment of inertia of the propeller about the axis of rotation and (c) the total kinetic energy of the propeller.

Salamat A.

Numerade Educator

Figure P8.76 shows a clawhammer as it is being used to pull a nail out of a horizontal board. If a force of magnitude 150 N is exerted horizontally as shown, find (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface at the point of contact with the hammer head. Assume that the force the hammer exerts on the nail is parallel to the nail and perpendicular to the position vector from the point of contact.

Averell H.

Carnegie Mellon University

A 40.0-kg child stands at one end of a 70.0-kg boat that is 4.00 m long (Fig. P8.77). The boat is initially 3.00 m from the pier. The child notices a turtle on a rock beyond the far end of the boat and proceeds to walk to that end to catch the turtle. (a) Neglecting friction between the boat and water, describe the motion of the system (child plus boat). (b) Where will the child be relative to the pier when he reaches the far end of the boat? (c) Will he catch the turtle? (Assume that he can reach out 1.00 m from the end of the boat.)

Salamat A.

Numerade Educator

An object of mass $M=$ 12.0$\quad \mathrm{kg}$ is attached to $\mathrm{a}$ cord that is wrapped around a wheel of radius $r=10.0 \mathrm{cm}(\mathrm{Fig.} \mathrm{P8.78})$ The acceleration of the object down the friction- less incline is measured to be $a=2.00 \mathrm{m} / \mathrm{s}^{2}$ and the incline makes an angle $\theta=37.0^{\circ}$ with the horizontal. Assuming the axle of the wheel to be frictionless, determine (a) the tension in the rope, (b) the moment of inertia of the wheel, and (c) the angular speed of the wheel 2.00 s after it begins rotating, starting from rest.

Averell H.

Carnegie Mellon University

A uniform ladder of length $L$ and weight $w$ is leaning against a vertical wall. The coefficient of static friction between the ladder and the floor is the same as that between the ladder and- the wall. If this coefficient of static friction is $\mu_{s}=0.500$ , determine the smallest angle the ladder can make with the floor-without slipping.

Salamat A.

Numerade Educator

Two astronauts (Fig. P8.80), each having a mass of 75.0 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 5.00 m/s. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum and (b) the rotational energy of the system. By pulling on the rope, the astronauts shorten the distance between them to 5.00 m. (c) What is the new angular momentum of the system? (d) What are their new speeds? (e) What is the new rotational energy of the system? (f) How much work is done by the astronauts in shortening the rope?

Averell H.

Carnegie Mellon University

This is a symbolic version of problem $80 .$ Two astronauts (Fig. P 8.80), each having a mass $M,$ are connected by a rope of length $d$ having negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed $v .$ (a) Calculate the magnitude of the angular momentum of the system by treating the astronauts as particles. (b) Calculate the rotational energy of the system. By pulling on the rope, the astronauts shorten the distance between them to d/2. (c) What is the new angular momentum of the system? (d) What are their new speeds? (e) What is the new rotational energy of the system? (f) How much work is done by the astronauts in shortening the rope?

Salamat A.

Numerade Educator

Two window washers, Bob and Joe, are on a 3.00-m-long, 345-N scaffold supported by two cables attached to its ends. Bob weighs 750 N and stands 1.00 m from the left end, as shown in Figure P 8.82. Two meters from the left end is the 500-N washing equipment. Joe is 0.500 m from the right end and weighs 1 000 N. Given that the scaffold is in rotational and translational equilibrium, what are the forces on each cable?

Averell H.

Carnegie Mellon University

A 2.35-kg uniform bar of length $\ell=1.30 \mathrm{m}$ is held in a horizontal position by three vertical springs as in Figure P8.83. The two lower springs are compressed and exert upward forces on the bar of magnitude $F_{1}=6.80 \mathrm{N}$ and $F_{2}=9.50 \mathrm{N},$ respectively. Find (a) the force $F s$ exerted by the top spring on the bar, and (b) the location x of the upper spring that will keep the bar in equilibrium.

Salamat A.

Numerade Educator

A light rod of length 2$L$ is free to rotate in a vertical plane about a frictionless pivot through its center. A particle of mass $m_{1}$ is attached at one end of the rod, and a mass $m_{2}$ is at the opposite end, where $m_{1}>m_{2}$ . The system is released from rest in the vertical position shown in Figure P8.84a, and at some later time, the system is rotating in the position shown in Figure P8.84b. Take the reference point of the gravitational potential energy to be at the pivot. (a) Find an expression for the system’s total mechanical energy in the vertical position. (b) Find an expression for the total mechanical energy in the rotated position shown in Figure P8.84b. (c) Using the fact that the mechanical energy of the system is conserved, how would you determine the angular speed v of the system in the rotated position? (d) Find the magnitude of the torque on the system in the vertical position and in the rotated position. Is the torque constant? Explain what these results imply regarding the angular momentum of the system. (e) Find an expression for the magnitude of the angular acceleration of the system in the rotated position. Does your result make sense when the rod is horizontal? When it is vertical? Explain.

Averell H.

Carnegie Mellon University

Many aspects of a gymnast’s motion can be modeled by representing the gymnast by four segments consisting of arms, torso (including the head), thighs, and lower legs, as in Figure $P 8.85 .$ Figure $P 8.85 b$ shows arrows of lengths $r_{\text { cg }}$ locating the center of gravity of each segment. Use the data below and the coordinate system shown in Figure P 8.85b to locate the center of gravity of the gymnast shown in Figure P 8.85a. Masses for the arms, thighs, and legs include both appendages.

Salamat A.

Numerade Educator

A uniform thin rod of length $L$ and mass $M$ is free to rotate on a frictionless pin passing through one end (Fig. P8.47). The rod is released from rest in the horizontal position. (a) What is the speed of its center of gravity when the rod reaches its lowest position? (b) What is the tangential speed of the lowest point on the rod when it is in the vertical position?

Averell H.

Carnegie Mellon University

A uniform solid cylinder of mass $M$ and radius $R$ rotates on a friction-less horizontal axle (Fig. P8.87). Two objects with equal masses $m$ hang from light cords wrapped around the cylinder. If the system is released from rest, find (a) the tension in each cord and (b) the acceleration of each object after the objects have descended a distance $h .$

Salamat A.

Numerade Educator

A painter climbs a ladder leaning against a smooth wall. At a certain height, the ladder is on the verge of slipping. (a) Explain why the force exerted by the vertical wall on the ladder is horizontal. (b) If the ladder of length L leans at an angle u with the horizontal, what is the lever arm for this horizontal force with the axis of rotation taken at the base of the ladder? (c) If the ladder is uniform, what is the lever arm for the force of gravity acting on the ladder? (d) Let the mass of the painter be $80 \mathrm{kg}, L=4.0 \mathrm{m},$ the ladder's mass be 30 $\mathrm{kg}$ , $\theta=53^{\circ},$ and the coefficient of friction between ground and ladder be $0.45 .$ Find the maximum distance the painter can climb un the ladder.

Averell H.

Carnegie Mellon University

A war-wolf, or trebuchet, is a device used during the Middle Ages to throw rocks at castles and now sometimes used to fling pumpkins and pianos. A simple trebuchet is shown in Figure P8.89. Model it as a stiff rod of negligible mass 3.00 m long and joining particles of mass $m_{1}=0.120 \mathrm{kg}$ and $m_{2}=$ 60.0 $\mathrm{kg}$ at its ends. It can turn on a frictionless horizontal axle perpendicular to the rod and 14.0 $\mathrm{cm}$ from the particle of larger mass. The rod is released from rest in a horizontal orientation. Find the maximum speed that the object of smaller mass attains.

Salamat A.

Numerade Educator

A string is wrapped around a uniform cylinder of mass $M$ and radius $R$ . The cylinder is released from rest with the string vertical and its top end tied to a fixed bar (Fig. $P 8.90 )$ . Show that (a) the tension in the string is one-third the weight of the cylinder, (b) the magnitude of the acceleration of the center of gravity is $2 g / 3,$ and $(c)$ the speed of the center of gravity is $(4 g h / 3)^{1 / 2}$ after the cylinder has descended through distance $h .$ Verify your answer to part $(\mathrm{c})$ with the energy approach.

Averell H.

Carnegie Mellon University

The Iron Cross When a gymnast weighing 750 N executes the iron cross as in Figure P8.91a, the primary muscles involved in supporting this position are the latissimus dorsi (“lats”) and the pectoralis major (“pecs”). The rings exert an upward force on the arms and support the weight of the gymnast. The force exerted by the shoulder joint on the arm is labeled $\overrightarrow{\mathbf{F}},$ while the two muscles exert a total force $\overrightarrow{\mathbf{F}}_{m}$ on the arm. Estimate the magnitude of the force $\overrightarrow{\mathbf{F}}_{m}$ . Note that one ring supports half the weight of the gymnast, which is 375 $\mathrm{N}$ as indicated in Figure $\mathrm{P} 8.91 \mathrm{b}$ . Assume that the force $\overrightarrow{\mathbf{F}}_{m}$ acts at an angle of $45^{\circ}$ below the horizontal at a distance of 4.0 $\mathrm{cm}$ from the shoulder joint. In your estimate, take the distance from the shoulder joint to the hand to be $L=70 \mathrm{cm}$ and ignore the weight of the arm.

Salamat A.

Numerade Educator

In an emergency situation, a person with a broken forearm ties a strap from his hand to clip on his shoulder as in Figure P8.92. His 1.60-kg forearm remains in a horizontal position and the strap makes an angle of $\theta=50.0^{\circ}$ with the horizontal. Assume the forearm is uniform, has a length of $\ell=0.320 \mathrm{m},$ assume the biceps muscle is relaxed, and ignore the mass and length of the hand. Find (a) the tension in the strap and (b) the components of the reaction force exerted by the humerus on the forearm.

Averell H.

Carnegie Mellon University

An object of mass $m_{1}=4.00 \mathrm{kg}$ is connected by a light cord to an object of mass $m_{2}=3.00 \mathrm{kg}$ on a frictionless surface (Fig. P8.93) The pulley rotates about a friction-less axle and has a moment of inertia of 0.500 $\mathrm{kg} \cdot \mathrm{m}^{2}$ and a radius of 0.300 $\mathrm{m} .$ Assuming that the cord does not slip on the pulley, find (a) the acceleration of the two masses and (b) the tensions $T_{1}$ and $T_{2}$ .

Salamat A.

Numerade Educator

A 10.0 -kg monkey climbs a uniform ladder with weight $w=$ $1.20 \times 10^{2} \mathrm{N}$ and length $L=3.00 \mathrm{m}$ as shown in Figure $\mathrm{P8.94}$ . The ladder rests against the wall at an angle of $\theta=60.0^{\circ} .$ The upper and lower ends of the ladder rest on friction-less surfaces, with the lower end fastened to the wall by a horizontal rope that is frayed and that can support a maximum tension of only 80.0 N. (a) Draw a force diagram for the ladder. (b) Find the normal force exerted by the bottom of the ladder. (c) Find the tension in the rope when the monkey is two-thirds of the way up the ladder. (d) Find the maximum distance $d$ that the monkey can climb up the ladder before the rope breaks. (e) If the horizontal surface were rough and the rope were removed, how would your analysis of the problem be changed and what other information would you need to answer parts (c) and (d)?

Averell H.

Carnegie Mellon University

A 3.2-kg sphere is suspended by a cord that passes over a 1.8-kg pulley of radius 3.8 cm. The cord

is attached to a spring whose force constant is $k=86 \mathrm{N} / \mathrm{m}$ as in Figure $\mathrm{P8.95}$ . Assume the pulley is a solid disk. (a) If the sphere is released from rest with the spring unstretched, what distance does the sphere fall through before stopping? (b) Find the speed of the sphere after it has fallen 25 cm.

Salamat A.

Numerade Educator