College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 8

Rotational Equilibrium and Rotational Dynamics - all with Video Answers

Educators

Chapter Questions

Problem 1

A grinding wheel of radius $0.350 \mathrm{~m}$ rotating on a frictionless axle is brought to rest by applying a constant friction force tangential to its rim. The constant torque produced by this force is $76.0 \mathrm{~N} \cdot \mathrm{m}$. Find the magnitude of the friction force.

Donald Albin

Donald Albin

Numerade Educator

Problem 2

According to the manual of a certain car, a maximum torque of magnitude $65.0 \mathrm{~N} \cdot \mathrm{m}$ should be applied when tightening the lug nuts on the vehicle. If you use a wrench of length $0.330 \mathrm{~m}$ and you apply the force at the end of the wrench at an angle of $75.0^{\circ}$ with respect to a line going from the lug nut through the end of the handle, what is the magnitude of the maximum force you can exert on the handle without excceding the recommendation?

Donald Albin

Numerade Educator

Problem 3

Calculate the net torque (magnitude and direction) on the beam in Figure $\mathrm{P} 8.8$ about (a) an axis through $O$ perpendicular to the page and (b) an axis through Cperpendicular to the page.

Donald Albin

Numerade Educator

Problem 4

A steel band exerts a horizontal force of $80.0 \mathrm{~N}$ on a tooth at point $B$ in Figure $P 8.4$. What is the torque on the root of the tooth about point $A$ ?

Donald Albin

Numerade Educator

Problem 5

A simple pendulum consists of a small object of mass $3.0 \mathrm{~kg}$ 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 $5.0^{\circ}$ angle with the vertical. (b) Does the torque increase or decrease af the angle increases? Explain.

Donald Albin

Numerade Educator

Problem 6

Write the necessary equations of equilibrium of the object shown in Figure P8.6. Take the origin of the torque equation about an axis perpendicular to the page through the point $O$.

Averell Hause

Carnegie Mellon University

Problem 7

The arm in Figure $\mathrm{P} 8.7$ weighs $41.5 \mathrm{~N}$. The force of gravity acting on the arm acts through point A. Determine the magnitudes of the tension force $\overrightarrow{\mathbf{F}}$, in the deltoid muscle and the force $\overrightarrow{\mathbf{F}}_{1}$ exerted by the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown.

Donald Albin

Numerade Educator

Problem 8

A uniform beam of length $7.60 \mathrm{~m}$ and weight $4.50 \times 10^{2} \mathrm{~N}$ is carried by two workers, Sam and Joe, as shown in Figure P8.8. (a) Determine the forces that each person exerts on the beam. (b) Qualitatively, how would the answers change if Sam moved closer to the midpoint?
(c) What would happen if Sam moved beyond the midpoint?

Donald Albin

Numerade Educator

Problem 9

A cook holds a $2.00-\mathrm{kg}$ carton of milk at arm's length (Fig. P8.9). What force $\overrightarrow{\mathbf{F}}_{n}$ must be exerted by the biceps muscle? (Ignore the weight of the forearm.)

Donald Albin

Numerade Educator

Problem 10

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

Donald Albin

Numerade Educator

Problem 11

Find the $x$ -and y-coordinates of the center of gravity of $a$ $4.00-f t$ by $8.00-f$ uniform sheet of plywood with the upper right quadrant removed as shown in Figure $\mathrm{P} 8.11 .$

Donald Albin

Numerade Educator

Problem 12

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 $\mathrm{P} 8.12 .$ 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) Usc 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?

Donald Albin

Numerade Educator

Problem 13

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

Donald Albin

Numerade Educator

Problem 14

A beam of length $L$ and mass $M$ rests on two pivots. The first pivot is at the left end, taken as the origin, and the second pivot is at a distance $\ell$ from the left end. $A$ woman of mass un starts at the left end and walks toward the right end as in Figure P8.12. When the beam is on the
verge of tipping, find symbolic expressions for (a) the normal force exerted by the second pivot in terms of $M, m$, and $g$ and (b) the woman's position in terms of $M, m, L$, and $\ell$. (c) Find the minimum value of $\ell$ that will allow the woman to reach the end of the beam without it tipping.

Donald Albin

Numerade Educator

Problem 15

Many of the elements in horizontal-bar exercises can be modeled by representing the gymnast by four segments consisting of arms, torso (including the head), thighs, and lower legs, as shown in Figure P8.15a. Inertial parameters for a particular gymnast are as follows:
Note that in Figure P8.15a $r_{\mathrm{cg}}$ is the distance to the center of gravity measured from the joint closest to the bar and the masses for the arms, thighs, and legs include both appendages. $I$ is the moment of inertia of each segment about its center of gravity. Determine the distance from the bar to the center of gravity of the gymnast for the two positions shown in Figures $\mathrm{P} 8.15 \mathrm{~b}$ and $\mathrm{P8}, 15 \mathrm{c}$.

Donald Albin

Numerade Educator

Problem 16

Using the data given in Problem 15 and the coordinate system shown in Figure $\mathrm{P} 8.16 \mathrm{~b}$, calculate the position of the center of gravity of the gymnast shown in Fig. ure P8.16a. Pay close attention to the definition of $r_{c}$ in the table.

Salamat Ali

Numerade Educator

Problem 17

A person bending forward to lift a load "with his back" (Fig. P8.17a) 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 P8.17b 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 $350 \mathrm{~N}$, 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 $12.0^{\circ}$. Find the tension in the back muscle and the compressional force in the spine.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 18

When a person stands on tiptoe (a strenuous position), the position of the foot is as shown in Figure $\mathrm{P} 8.18 \mathrm{a}$. The total gravitational force on the body, $\overrightarrow{\mathbf{F}}_{k}$, is supported by the force $\vec{n}$ exerted by the floor on the toes of one foot. A mechanical model of the situation is shown in Figure P8.18b, 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 $F_{g}=700 \mathrm{~N}$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 19

A $500-\mathrm{N}$ uniform rectangular sign $4.00 \mathrm{~m}$ wide and $3.00 \mathrm{~m}$ high is suspended from a horizontal, $6.00$ -m-long, uniform, $100-\mathrm{N}$ rod as indicated in Figure $\mathrm{P} 8.19 .$ The left end of the rod is supported by a hinge, and the right end is supported by a thin cable making a $30.0^{\circ}$ 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.

Donald Albin

Numerade Educator

Problem 20

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

Averell Hause

Carnegie Mellon University

Problem 21

A uniform plank of length $2.00 \mathrm{~m}$ and mass $30.0 \mathrm{~kg}$ is supported by three ropes, as indicated by the blue vectors in
Figure $\mathrm{P} 8.21 .$ Find the tension in each rope when a $700-\mathrm{N}$ person is $0.500 \mathrm{~m}$ from the left end.

Donald Albin

Numerade Educator

Problem 22

A hungry $700-\mathrm{N}$ bear walks out on a beam in an attempt to retrieve some "goodies" hanging at the end (Fig. P8.22). The beam is uniform, weighs $200 \mathrm{~N}$, and is $6.00 \mathrm{~m}$ long; the goodies weigh $80.0 \mathrm{~N}$. (a) Draw a free-body diagram of the beam. (b) When the bear is at $x=1.00 \mathrm{~m}$, find the tension in the wire and the components
of the reaction force at the hinge. (c) If the wire can withstand a maximum tension of $900 \mathrm{~N}$, what is the maximum distance the bear can walk before the wire breaks?

Donald Albin

Numerade Educator

Problem 23

An $8.00-\mathrm{m}, 200-\mathrm{N}$ uniform ladder rests against a smooth wall. The coefficient of static friction between the ladder and the ground is $0.600$, and the ladder makes a $50.0^{\circ}$ angle with the ground. How far up the ladder can an $800-\mathrm{N}$ person climb before the ladder begins to slip?

Donald Albin

Numerade Educator

Problem 24

A strut of length $L=3.00 \mathrm{~m}$ and mass $m=16.0 \mathrm{~kg}$ is held by a cable at an angle of $\theta=30.0^{\circ}$ with respect to the horizontal as shown in Figure $\mathrm{P8} .24$. (a) Sketch a free-body 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.

Donald Albin

Numerade Educator

Problem 25

A student gets his car stuck in a snowdrift. Not at a loss, having studied physics, he attaches one end of a stout rope to the car and the other end to the trunk of a nearby tree, allowing for a small amount of slack. The student then exerts a force $\overrightarrow{\mathbf{F}}$ on the center of the rope in the direction perpendicular to the car-tree line as shown in Figure $\mathrm{P} 8.25 .$ If the rope is inextensible and the magnitude of the applied force is $475 \mathrm{~N}$, what is the force on the car? (Assume equilibrium conditions.)

Donald Albin

Numerade Educator

Problem 26

A uniform beam of length $L$ and mass $m$ shown in Figure $\mathrm{P} 8.26$ is inclined at an angle $\theta$ to the horizontal. Its upper end is connected to a wall by a rope, and its lower end rests on a rough horizontal surface. The coefficient of static friction between the beam and
surface is $\mu_{s^{\prime}}$ Assume the angle is such that the static friction force is at its maximum value. (a) Draw a free-body diagram for the beam. (b) Using the condition of rotational equilibrium, find an expression for the tension $T$ in the rope in terms of $m, g$, and $\theta .$ (c) Using Newton's second law for equilibrium, find a second expression for Tin terms of $\mu_{s}, m$, and $g$. (d) Using the foregoing results, obtain a relationship involving only $\mu_{s}$ and the angle $\theta$.
(e) What happens if the angle gets smaller? Is this equation valid for all values of $\theta$ ? Explain.

Donald Albin

Numerade Educator

Problem 27

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 P8.27a. 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 $\mathrm{P} 8.27 \mathrm{~b}, \overrightarrow{\mathbf{F}}_{c}$ is the force exerted by the food being chewed against the jawbone, $\overrightarrow{\mathrm{T}}$ is the force of tension in the masseter, and $\overrightarrow{\mathbf{R}}$ is the force exerted by the socket on the mandible. Find $\vec{T}$ and $\vec{R}$ for a person who bites down on a piece of steak with a force of $50.0 \mathrm{~N}$.

Salamat Ali

Numerade Educator

Problem 28

A $1200-\mathrm{N}$ uniform boom is supported by a cable perpendicular to the boom as in Figure $\mathrm{P} 8.28$. The boom is hinged at the bottom, and a $2000-\mathrm{N}$ weight hangs from its top. Find the tension in the supporting cable and the components of the reaction force exerted on the boom by the hinge.

Donald Albin

Numerade Educator

Problem 29

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. $\mathrm{P} 8.29 \mathrm{a}$ ). The forces on the lower leg when the leg is extended are modeled as in Figure $\mathrm{P} 8.29 \mathrm{~b}$, where $\vec{T}$ 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 $\vec{T}$ when the tendon is at an angle of $25.0^{\circ}$ 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.

Donald Albin

Numerade Educator

Problem 30

One end of a uniform $4.0-\mathrm{m}$ long rod of weight $w$ is supported by a cable. The other end rests against a wall, where it is held by friction. (See Fig. P8.30.) The coefficient of static friction between the wall and the rod is $\mu_{\mathrm{s}}=0.50 .$ 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 $A$.

Donald Albin

Numerade Educator

Problem 31

Four objects are held in position at the corners of a rectangle by light rods as shown in Figure $\mathrm{P} 8.31$. Find the moment of inertia of the system about (a) the $x$ -axis.
(b) the y-axis, and (c) an axis through $O$ and perpendicular to the page.
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?

Donald Albin

Numerade Educator

Problem 32

If the system shown in Figure $\mathrm{P} 8.31$ is set in rotation about each of the axes mentioned in Problem 30 , find the torque that will produce an angular acceleration of $1.50 \mathrm{rad} / \mathrm{s}^{2}$ in each case.

Donald Albin

Numerade Educator

Problem 33

A large grinding wheel in the shape of a solid cylinder of radius $0.330 \mathrm{~m}$ is free to rotate on a frictionless, vertical axle. A constant tangential force of $250 \mathrm{~N}$ applied to its edge causes the wheel to have an angular acceleration of $0.940 \mathrm{rad} / \mathrm{s}^{2}$. (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 $5.00 \mathrm{~s}$ have elapsed, assuming the force is acting during that time?

Salamat Ali

Numerade Educator

Problem 34

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 $\mathrm{P} 8.34 .$
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 $\mathrm{P} 8.34$, 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 $\alpha$ 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 $g .$ (f) Write Newton's second law for rotation in terms of $I, \alpha, T$, and $r$. (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?

Donald Albin

Numerade Educator

Problem 35

A rope of negligible mass is wrapped around a $225-\mathrm{kg}$ solid cylinder of radius $0.400 \mathrm{~m} .$ The cylinder is suspended several meters off the ground with its axis oriented horizontally, and turns on that axis without friction. (a) If a $75.0-\mathrm{kg}$ man takes hold of the free end of the rope and falls under the force of gravity, what is his acceleration? (b) What is the angular acceleration of the cylinder? (c) If the mass of the rope were not neglected, what would happen to the angular acceleration of the cylinder as the man falls?

Donald Albin

Numerade Educator

Problem 36

A potter's wheel having a radius of $0.50 \mathrm{~m}$ 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 $6.0 \mathrm{~s}$ by pressing a wet rag against the rim and exerting a radially inward force of $70 \mathrm{~N}$. Find the effective coefficient of kinetic friction between the wheel and the wet rag.

Donald Albin

Numerade Educator

Problem 37

A model airplane with mass $0.750 \mathrm{~kg}$ is tethered by a wire so that it flies in a circle $30.0 \mathrm{~m}$ in radius. The airplane engine provides a net thrust of $0.800 \mathrm{~N}$ 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 Ali

Numerade Educator

Problem 38

$A$ bicycle wheel has a diameter of $64.0 \mathrm{~cm}$ and a mass of $1.80 \mathrm{~kg}$. 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 $120 \mathrm{~N}$ is applied tangent to the rim of the tire. (a) What force must be applied by a chain passing over a $9.00-\mathrm{cm}$ diameter sprocket in order to give the wheel an acceleration of $4.50 \mathrm{rad} / \mathrm{s}^{22} ?$ (b) What force is required if you shift to a $5.60-\mathrm{cm}-\mathrm{diameter}$ sprocket?

Donald Albin

Numerade Educator

Problem 39

A $150-\mathrm{kg}$ merry-go-round in the shape of a uniform, solid, horizontal disk of radius $1.50 \mathrm{~m}$ 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 $0.500 \mathrm{rev} / \mathrm{s}$ in $2.00 \mathrm{~s}$ ?

Salamat Ali

Numerade Educator

Problem 40

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 running over a pulley as in Figure $\mathrm{P} 8.40 .$ 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 $T_{2}$.

Donald Albin

Numerade Educator

Problem 41

An airliner lands with a speed of $50.0 \mathrm{~m} / \mathrm{s}$. Each wheel of the plane has a radius of $1.25 \mathrm{~m}$ and a moment of inertia of $110 \mathrm{~kg} \cdot \mathrm{m}^{2}$. At touchdown, the wheels begin to spin under the action of friction. Each wheel supports a weight of $1.40 \times 10^{4} \mathrm{~N}$, and the wheels attain their angular speed in $0.480 \mathrm{~s}$ while rolling without slipping. What is the coefficient of kinetic friction between the wheels and the runway? Assume that the speed of the plane is constant.

Donald Albin

Numerade Educator

Problem 42

A car is designed to get its energy from a rotating flywheel with a radius of $2.00 \mathrm{~m}$ and a mass of $500 \mathrm{~kg}$. Before a trip, the flywheel is attached to an electric motor, which brings the flywheel's rotational speed up to $5000 \mathrm{rev} / \mathrm{min} .$
(a) Find the kinetic energy stored in the flywheel. (b) If the flywheel 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.

Donald Albin

Numerade Educator

Problem 43

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

Salamat Ali

Numerade Educator

Problem 44

Four objects -a hoop, a solid cylinder, a solid sphere, and a thin, spherical shell-each has a mass of $4.80 \mathrm{~kg}$ and a radius of $0.230 \mathrm{~m}$. (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.

Donald Albin

Numerade Educator

Problem 45

A light rod $1.00 \mathrm{~m}$ in length rotates about an axis perpendicular to its length and passing through its center as in Figure $\mathrm{P} 8.45$. Two particles of masses $4.00 \mathrm{~kg}$ and $3.00 \mathrm{~kg}$ 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 $2.50 \mathrm{rad} / \mathrm{s} ?$ (b) Repeat the problem, assuming the mass of the rod is taken to be $2.00 \mathrm{~kg}$,

Donald Albin

Numerade Educator

Problem 46

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

Donald Albin

Numerade Educator

Problem 47

A solid, uniform disk of radius $0.250 \mathrm{~m}$ and mass $55.0 \mathrm{~kg}$ rolls down a ramp of length $4.50 \mathrm{~m}$ that makes an angle of $15.0^{\circ}$ 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.

Donald Albin

Numerade Educator

Problem 48

A solid uniform sphere of mass $m$ and radius $R$ rolls without slipping down an incline of height $h$. (a) What forms of mechanical energy are associated with the sphere at any point along the incline when its angular speed is $\omega$ ? Answer in words and symbolically in terms of the quantities $m, g, y, I, \omega$, and $v$. (b) What force acting on the sphere causes it to roll rather than slip down the incline? (c) Determine the ratio of the sphere's rotational kinetic energy to its total kinetic energy at any instant.

Donald Albin

Numerade Educator

Problem 49

The top in Figure $\mathrm{P} 8.49$ 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 $A A^{\prime}$. 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 $5.57 \mathrm{~N}$ in the string. If the string does not slip while wound around the
peg, what is the angular speed of the top after $80.0 \mathrm{~cm}$ of string has been pulled off the peg? Hint: Consider the work that is done.

Donald Albin

Numerade Educator

Problem 50

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 $W_{\text {net }}=F \Delta x=\frac{1}{2} m u_{i}^{2}-\frac{1}{2} m v_{i}^{2}$ is $W_{\text {net }}$ $=\tau \Delta \theta=\frac{1}{2} I \omega_{f}^{2}-\frac{1}{2} I \omega_{i}^{2}$. You should convince yourself that
this relationship is correct.

Donald Albin

Numerade Educator

Problem 51

A $10.0-\mathrm{kg}$ cylinder rolls without slipping on a rough surface. At an instant when its center of gravity has a speed of $10.0 \mathrm{~m} / \mathrm{s}$, 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 Ali

Numerade Educator

Problem 52

Use conservation of energy to determine the angular speed of the spool shown in Figure $\mathrm{P} 8.52$ after the
$3.00-\mathrm{kg}$ bucket has fallen $4.00 \mathrm{~m}$, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds.

Averell Hause

Carnegie Mellon University

Problem 53

A giant swing at an amusement park consists of a $365-\mathrm{kg}$ uniform arm $10.0 \mathrm{~m}$ long, with two seats of negligible mass connected at the lower end of the arm (Fig. $\mathrm{P8} .53$ ). (a) How far from the upper end is the center of mass of the arm? (b) The gravitational potential energy of the arm is the same as if all its mass were concentrated at the center of mass. If the arm is raised through a $45.0^{\circ}$ angle, find the gravitational potential energy, where the zero level is taken to be $10.0 \mathrm{~m}$ below the axis. (c) The arm drops from rest from the position described in part
(b). Find the gravitational potential energy of the system when it reaches the vertical orientation. (d) Find the speed of the seats at the bottom of the swing.

Donald Albin

Numerade Educator

Problem 54

Each of the following objects has a radius of $0.180 \mathrm{~m}$ and a mass of $2,40 \mathrm{~kg}$, and each rotates about an axis through its center (as in Table 8.1) with an angular speed of $35.0 \mathrm{rad} / \mathrm{s}$. 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

Donald Albin

Numerade Educator

Problem 55

(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 Ali

Numerade Educator

Problem 56

a 0.005$ -kg bullet traveling horizontally with a speed of $1.00 \times 10^{3} \mathrm{~m} / \mathrm{s}$ enters an
$18.0-\mathrm{kg}$ door, imbedding itself $10.0 \mathrm{~cm}$ from the side opposite the hinges as in Figure $\mathrm{P} 8.56$. The $1.00-\mathrm{m}$ -wide door is free to swing on its hinges. (a) Before it hits the door, does the bullet have
angular momentum rela-
tive 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.

Donald Albin

Numerade Educator

Problem 57

A light rigid rod $1.00 \mathrm{~m}$ in length rotates about an axis perpendicular to its length and through its center, as shown in Figure P8.45. Two particles of masses $4.00 \mathrm{~kg}$ and $3.00 \mathrm{~kg}$ 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.)

Donald Albin

Numerade Educator

Problem 58

Halley's comet moves about the Sun in an elliptical orbit, with its closest approach to the Sun being $0.59 \mathrm{~A} . \mathrm{U} .$ and its greatest distance being $35 \mathrm{A.U.}(1 \mathrm{~A}, \mathrm{U}$, is the EarthSun distance). If the comet's speed at closest approach is $54 \mathrm{~km} / \mathrm{s}$, 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.

Donald Albin

Numerade Educator

Problem 59

The system of small objects shown in Figure $\mathrm{P} 8.59$ is rotating at an angular spced of $2.0 \mathrm{rev} / \mathrm{s} .$ The objects are connected by light, flexible spokes that can be lengthened or shortened. What is the new angular speed if the spokes are shortened to $0.50 \mathrm{~m}$ ? (An effect similar to that illustrated in this problem occurred in the early stages of the formation of our galaxy. As the massive cloud of dust and gas that was the source of the stars and planets contracted, an initially small angular speed increased with time.)

Donald Albin

Numerade Educator

Problem 60

A playground merry-go-round of radius $2.00 \mathrm{~m}$ has a moment of inertia $I=275 \mathrm{~kg} \cdot \mathrm{m}^{2}$ and is rotating about a frictionless vertical axle. As a child of mass $25.0 \mathrm{~kg}$ stands at a distance of $1.00 \mathrm{~m}$ from the axle, the system (merrygo-round and child) rotates at the rate of $14.0 \mathrm{rev} / \mathrm{min}$. The child then proceeds to walk toward the edge of the merry-go-round. What is the angular speed of the system when the child reaches the edge?

Donald Albin

Numerade Educator

Problem 61

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

Salamat Ali

Numerade Educator

Problem 62

A student sits on a rotating stool holding two $3.0-\mathrm{kg}$ objects. When his arms are extended horizontally, the objects are $1.0 \mathrm{~m}$ from the axis of rotation and he rotates with an angular speed of $0.75 \mathrm{rad} / \mathrm{s}$. 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 Hause

Carnegie Mellon University

Problem 63

The puck in Figure $\mathrm{P8} .63$ has a mass of $0.120 \mathrm{~kg}$. Its original distance from the center of rotation is $40.0 \mathrm{~cm}$, and it moves with a speed of $80.0 \mathrm{~cm} / \mathrm{s}$. The string is pulled downward $15.0 \mathrm{~cm}$ 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 Ali

Numerade Educator

Problem 64

A space station shaped like a giant wheel has a radius of $100 \mathrm{~m}$ and a moment of inertia of $5.00 \times 10^{8} \mathrm{~kg} \cdot \mathrm{m}^{2}$. $\mathrm{A}$
crew of 150 lives on the rim, and the station is rotating so that the crew experiences an apparent acceleration of $1 g$ (Fig. P8.64). 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 $65.0 \mathrm{~kg}$.

Averell Hause

Carnegie Mellon University

Problem 65

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. P8.65). 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 Ali

Numerade Educator

Problem 66

A merry-go-round rotates at the rate of $0.20 \mathrm{rev} / \mathrm{s}$ with an 80 -kg man standing at a point $2.0 \mathrm{~m}$ from the axis of rotation. (a) What is the new angular speed when the man walks to a point $1.0 \mathrm{~m}$ from the center? Assume that the merry-go-round is a solid 25 -kg cylinder of radius $2.0 \mathrm{~m}$.
(b) Calculate the change in kinetic energy due to the man's movement. How do you account for this change in kinetic energy?

Donald Albin

Numerade Educator

Problem 67

A $60.0-\mathrm{kg}$ 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 $2.00 \mathrm{~m}$. 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 $1.50 \mathrm{~m} / \mathrm{s}$ 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 Hause

Carnegie Mellon University

Problem 68

Figure $\mathrm{P} 8.68$ shows a clawhammer as it is being used to pull a nail out of a horizontal board. If a force of magnitude $150 \mathrm{~N}$ 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.

Donald Albin

Numerade Educator

Problem 69

A $40.0$ -kg child stands at one end of a $70.0$ -kg boat that is $4.00 \mathrm{~m}$ long (Fig. P8.69). The boat is initially $3.00 \mathrm{~m}$ 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 $1.00 \mathrm{~m}$ from the end of the boat.)

Salamat Ali

Numerade Educator

Problem 70

A $12.0-\mathrm{kg}$ object is attached to a cord that is wrapped around a wheel of radius $r=10.0 \mathrm{~cm}$ (Fig. P8.70). The acceleration of the object down the frictionless incline is measured to be $2.00 \mathrm{~m} / \mathrm{s}^{2}$. 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 $2.00 \mathrm{~s}$ after it begins rotating, starting from rest.

Donald Albin

Numerade Educator

Problem 71

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 Ali

Numerade Educator

Problem 72

Two astronauts (Fig. P8.72), each having a mass of $75.0 \mathrm{~kg}$, are connected by a $10.0-\mathrm{m}$ rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of $5.00 \mathrm{~m} / \mathrm{s}$. 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 $5.00 \mathrm{~m}$. (c) What is the new angular momentum of the system? (d) What are (heir 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?

Donald Albin

Numerade Educator

Problem 73

Two astronauts (Fig. P8.72), 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 $y$.
(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?

Donald Albin

Numerade Educator

Problem 74

Two window washers, Bob and Joe, are on a $3.00$ -m-long, $345-\mathrm{N}$ scaffold supported by two cables attached to its ends. Bob weighs $750 \mathrm{~N}$ and stands $1.00 \mathrm{~m}$ from the left end, as shown in Figure $\mathrm{P} 8.74 .$ Two meters from the left end is the $500-\mathrm{N}$ washing equipment. Joe is $0.500 \mathrm{~m}$ from the right end and weighs $1000 \mathrm{~N}$. Given that the scaffold is in rotational and translational equilibrium, what are the forces on each cable?

Donald Albin

Numerade Educator

Problem 75

A star with mass $3.00 \times 10^{\text {sit }} \mathrm{kg}$ and radius $1.50 \times 10^{9} \mathrm{~m}$ rotates on its axis at a rate of $0.0100 \mathrm{rev} / \mathrm{d}$. If the star suddenly collapses to a neutron star of radius $15.0 \mathrm{~km}$, find (a) the angular speed of the star and (b) the tangential speed of an indestructible astronaut standing on the equator.

Donald Albin

Numerade Educator

Problem 76

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 $\mathrm{P} 8.76 \mathrm{a}$, and at some later time the system is rotating in the position shown in Figure $\mathrm{P} 8.76 \mathrm{~b}$. 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 $\mathrm{P} 8.76 \mathrm{~b}$. (c) Using the fact. that the mechanical energy of the system is conserved, how would you determine the angular speed $\omega$ 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.

Donald Albin

Numerade Educator

Problem 77

In Figure $\mathrm{P} 8.77$, the sliding block has a mass of $0.850 \mathrm{~kg}$, the counterweight has a mass of $0.420 \mathrm{~kg}$, and the pulley is a uniform solid cylinder with a mass of $0.350 \mathrm{~kg}$ and an outer radius of $0.0300 \mathrm{~m}$. The coefficient of kinctic friction between the block and the horizontal surface is $0.250 .$ The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pulley. The block has a velocity of $0.820 \mathrm{~m} / \mathrm{s}$ toward the pulley when it passes through a photogate. (a) Use energy methods to predict the speed of the block after it has moved to a second photogate $0.700 \mathrm{~m}$ away. (b) Find the angular speed of the pulley at the same moment.

Donald Albin

Numerade Educator

Problem 78

(a) Without the wheels, a bicycle frame has a mass of $8.44 \mathrm{~kg}$. Each of the wheels can be roughly modeled as a uniform solid disk with a mass of $0.820 \mathrm{~kg}$ and a radius of $0.349 \mathrm{~m}$. Find the kinetic energy of the whole bicycle when it is moving forward at $3.35 \mathrm{~m} / \mathrm{s}$. (b) Before the invention of a wheel turning on an axle, ancient people moved heavy loads by placing rollers under them. (Modern people use rollers, too: Any hardware store will sell you a roller bearing for a lazy Susan.) A stone block of mass $844 \mathrm{~kg}$ moves forward at $0.335 \mathrm{~m} / \mathrm{s}$, supported by two uniform cylindrical tree trunks, each of mass $82.0 \mathrm{~kg}$ and radius $0.343 \mathrm{~m}$. There is no slipping between the block and the rollers on between the rollers and the ground. Find the total kinetic energy of the moving objects.

Donald Albin

Numerade Educator

Problem 79

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 $\mathrm{P} 8.79$. A light plank rests on two scales that read $F_{\mathrm{gl}}=380 \mathrm{~N}$ and $F_{\mathrm{g} 2}=320 \mathrm{~N}$. The scales are separated by a distance of $2.00 \mathrm{~m}$. How far from the woman's feet is her center of gravity?

Donald Albin

Numerade Educator

Problem 80

In a circus performance, a large $5.0-\mathrm{kg}$ hoop of radius $3.0 \mathrm{~m}$ rolls without slipping. If the hoop is given an angular speed of $3.0 \mathrm{rad} / \mathrm{s}$ while rolling on the horizontal ground and is then allowed to roll up a ramp inclined at $20^{\circ}$ with the horizontal. how far along the incline does the hoon moll?

Donald Albin

Numerade Educator

Problem 81

A uniform solid cylinder of mass $M$ and radius $R$ rotates on a frictionless horizontal axle (Fig. P8.81). 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 $h .$

Donald Albin

Numerade Educator

Problem 82

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 $\theta$ 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 up the ladder.

Donald Albin

Numerade Educator

Problem 83

A mar-uolf, 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 P8.83. Model it as a stiff rod of negligible mass $3.00 \mathrm{~m}$ long and joining particles of mass $60.0 \mathrm{~kg}$ and $0.120 \mathrm{~kg}$ 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.

Donald Albin

Numerade Educator

Problem 84

$\mathrm{A}$ 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. P8.84). 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 (c) with the energy approach.

Averell Hause

Carnegie Mellon University

Problem 85

The Iron Cross When a gymnast weighing $750 \bar{N}$ executes the iron cross as in Figure $\mathrm{P} 8.85 \mathrm{a}$, 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 $\overrightarrow{\mathbf{F}}_{\mathrm{m}}$ 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 $\mathrm{P8} .85 \mathrm{~b}$. Assume that the force $\overrightarrow{\mathbf{F}}_{\mathrm{w}}$ 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 $70 \mathrm{~cm}$ and ignore the weight of the arm.

Donald Albin

Numerade Educator

Problem 86

Swinging on a high bar The gymnast shown in Figure P8. 86 is performing a backwards giant swing on the high bar. Starting from rest in a near-vertical orientation, he rotates around the bar in a counterclockwise direction. keeping his body and arms straight. Friction between the bar and the gymnast's hands exerts a constant torque opposing the rotational motion. If the angular velocity of the gymnast at position 2 is measured to be $4.0 \mathrm{rad} / \mathrm{s}$, determine his angular velocity at position 3. (Note that this maneuver is called a backwards giant swing, even though the motion of the gymnast would seem to be forwards.)

Donald Albin

Numerade Educator

Problem 87

A $4.00-\mathrm{kg}$ mass is connected by a light cord to a $3.00-\mathrm{kg}$ mass on a smooth surface (Fig. P8.87). The pulley rotates about a frictionless 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 $T_{2}$.

Donald Albin

Numerade Educator

Problem 88

A $10.0-\mathrm{kg}$ 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 $\mathrm{P} 8.88 .$ The ladder rests against the wall at an angle of $\theta=60.0^{\circ}$. The upper and lower ends of the ladder rest on frictionless surfaces, with the lower end fastened to the wall by a horizontal rope that is frayed and that can sup-
port a maximum tension of only $80.0 \mathrm{~N}$. (a) Draw a freebody 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.
(c) 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)?

Donald Albin

Numerade Educator