Essential University Physics, 3e (Wolfson) 3e (Wolfson) Chapter 10 Rotational Motion
1 0. 1 Co nc e pt ua lQue s t i o ns 1) When a rigid body rotates about a fixed axis, all the points in the body b ody have the same A) tangential speed. B) angular acceleration. C) tangential acceleration. D) linear displacement. ) centripetal acceleration. Ans!er" B #ar" 1
$) A hori%ontal dis& rotates about a vertical axis through its center. 'oint P 'oint P is is mid!ay bet!een the center and the rim of the dis&, and point Q is on the rim. (f the dis& turns !ith constant angular velocity, !hich of the follo!ing statements about it are true *+here may be more than one correct choice.) A) P A) P and and Q have the same linear acceleration. B) Q is moving t!ice as fast as P as P . C) +he linear acceleration of Q is t!ice as great as the linear acceleration of P of P . D) +he linear acceleration of P of P is is t!ice as great as the linear acceleration of Q. ) +he angular velocity of Q is Q is t!ice as great as the angular velocity of P P . Ans!er" B, C #ar" 1
) A dumbbell-shaped obect is composed by t!o e/ual masses, m, connected by a rod of negligible mass and length r. (f I (f I 1 is the moment of inertia of this obect !ith respect to an axis passing through the center of the rod and perpendicular to it and I and I $ is the moment of inertia !ith respect to an axis passing through one of the masses, it follo!s that A) I A) I 1 0 I 0 I $. B) I B) I 1 I I $. C) I C) I $ I I 1. Ans!er" C #ar" 1
2) (f t!o forces of e/ual magnitude act on an obect that is hinged at a pivot, the force acting farther from the pivot must produce the greater tor/ue about the pivot. A) true B) false C) unable to decide !ithout &no!ing the shape of the obect Ans!er" B #ar" 1
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6) +he t!o rotating systems sho!n in the figure differ only in that the t!o identical movable masses are positioned at different distances from the axis of rotation. (f you release the hanging bloc&s simultaneously from rest, and if the ropes do not slip, !hich bloc& lands first
A) +he bloc& at the left lands first. B) +he bloc& at the right lands first. C) Both bloc&s land at the same time. Ans!er" B #ar" 1
5) (f an irregularly shaped obect *such as a !rench) is dropped from rest in a classroom and feels no air resistance, it !ill A) accelerate but !ill not spin. B) accelerate and turn until its center of gravity reaches its lo!est point. C) accelerate and turn about its center of gravity !ith uniform angular speed. D) accelerate and turn about its center of gravity !ith uniform angular acceleration. ) accelerate and spin until its center of gravity reaches its highest point. Ans!er" A #ar" 1
7) A tire is rolling along a road, !ithout slipping, !ith a velocity v. A piece of tape is attached to the tire. When the tape is opposite the road *at the top of the tire), its velocity !ith respect to the road is A) $v. B) v. C) 1.6v. D) %ero. ) +he velocity depends on the radius of the tire. Ans!er" A #ar" 1
8) Consider a uniform solid sphere of radius R and mass M rolling !ithout slipping. Which form of its &inetic energy is larger, translational or rotational A) (ts translational &inetic energy is larger than its rotational &inetic energy. B) (ts rotational &inetic energy is larger than its translational &inetic energy. C) Both forms of energy are e/ual. D) 9ou need to &no! the speed of the sphere to tell. Ans!er" A #ar" 1
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:) A solid sphere, solid cylinder, and a hollo! pipe all have e/ual masses and radii and are of uniform density. (f the three are released simultaneously at the top of an inclined plane and roll !ithout slipping, !hich one !ill reach the bottom first A) solid sphere B) hollo! pipe C) solid cylinder D) +hey all reach the bottom at the same time. Ans!er" A #ar" 1
14) A uniform dis&, a uniform hoop, and a uniform solid sphere are released at the same time at the top of an inclined ramp. +hey all roll !ithout slipping. (n !hat order do they reach the bottom of the ramp A) dis&, hoop, sphere B) hoop, sphere, dis& C) sphere, dis&, hoop D) sphere, hoop, dis& ) hoop, dis&, sphere Ans!er" C #ar" 1
11) A ball is released from rest on a no-slip surface, as sho!n in the figure. After reaching its lo!est point, the ball begins to rise again, this time on a frictionless surface as sho!n in the figure. When the ball reaches its maximum height on the frictionless surface, it is
A) at a greater height as !hen it !as released. B) at a lesser height as !hen it !as released. C) at the same height as !hen it !as released. D) (t is impossible to tell !ithout &no!ing the mass of the ball. ) (t is impossible to tell !ithout &no!ing the radius of the ball. Ans!er" B #ar" 1
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1 0 . 2 Pr o bl e ms 1) A turbine blade rotates !ith angular velocity ;*t ) 0 $.44 rad
. What is the
angular acceleration of the blade at A) -8.$ rad< B) -1:.1 rad< C) -85.4 rad< D) -5.$ rad< ) -17$ rad< Ans!er" A #ar" 64=
$) +he angular velocity of a 766-g !heel 16.4 cm in diameter is given by the e/uation ω*t ) 0 *$.44 rad
) +he angular acceleration of a !heel is given in the !heel starts from rest at A) 6.1 s B) 8.2 s C) 5.: s D) .5 s Ans!er" A
by
!here t is in seconds. (f
!hen is the next time the !heel is at rest
#ar" 5
2) A 1.16-&g grinding !heel $$.4 cm in diameter is spinning countercloc&!ise at a rate of $4.4 revolutions per second. When the po!er to the grinder is turned off, the grinding !heel slo!s !ith constant angular acceleration and ta&es 84.4 s to come to a rest. *a) What !as the angular acceleration *in rado! many revolutions did the !heel ma&e during the time it !as coming to rest Ans!er" *a) -1.67 rad
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6) A .26-&g centrifuge ta&es 144 s to spin up from rest to its final angular speed !ith constant angular acceleration. A point located 8.44 cm from the axis of rotation of the centrifuge moves !ith a speed of 164 mo! many revolutions does the centrifuge ma&e as it goes from rest to its final angular speed Ans!er" *a) 18.8 rad
5) When a $.76-&g fan, having blades 18.6 cm long, is turned off, its angular speed decreases uniformly from 14.4 rad
7) A 2.64-&g !heel that is 2.6 cm in diameter rotates through an angle of 1.8 rad as it slo!s do!n uniformly from $$.4 rad
8) A machinist turns the po!er on to a grinding !heel, !hich is at rest at time t 0 4.44 s. +he !heel accelerates uniformly for 14 s and reaches the operating angular velocity of +he !heel is run at that angular velocity for 7 s and then po!er is shut off. +he !heel decelerates uniformly at
until the !heel stops. (n this situation, the time interval of angular
deceleration *slo!ing do!n) is closest to" A) 17 s B) 16 s C) 1: s D) $1 s ) $ s Ans!er" A #ar" 64=
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:) (n the figure, point P is at rest !hen it is on the x-axis. +he linear speed of point P !hen it reaches the y-axis is closest to
A) 4.18 m
14) (n the figure, point P is at rest !hen it is on the x-axis. +he time t , !hen P returns to the original position on the x-axis, is closest to
A) 1 s. B) 18 s. C) $6 s. D) 6 s. ) 64 s. Ans!er" D #ar" 1
11) A 1.$6-&g ball begins rolling from rest !ith constant angular acceleration do!n a hill. (f it ta&es .54 s for it to ma&e the first complete revolution, ho! long !ill it ta&e to ma&e the next complete revolution Ans!er" 1.2: s #ar" 1
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1$) A piece of thin uniform !ire of mass m and length b is bent into an e/uilateral triangle. ind the moment of inertia of the !ire triangle about an axis perpendicular to the plane of the triangle and passing through one of its vertices. A)
mb$
B)
mb$
C)
mb$ mb$
D) )
mb$
Ans!er" #ar" 1
1) A slender uniform rod 144.44 cm long is used as a meter stic&. +!o parallel axes that are perpendicular to the rod are considered. +he first axis passes through the 64-cm mar& and the second axis passes through the 4-cm mar&. What is the ratio of the moment of inertia through the second axis to the moment of inertia through the first axis A) I $< I 1 0 1.6 B) I $< I 1 0 1.7 C) I $< I 1 0 1.: D) I $< I 1 0 $.1 ) I $< I 1 0 $. Ans!er" A #ar" 1
12) A uniform solid sphere has a moment of inertia I about an axis tangent to its surface. What is the moment of inertia of this sphere about an axis through its center A) 1<7 I B) $<7 I C) $<6 I D) <6 I ) 7<6 I Ans!er" B #ar" 1
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16) (n the figure, a !eightlifters barbell consists of t!o identical uniform spherical masses each !ith radius 4.17 m and mass of 64 &g. +he !eights are connected by a 4.:5-m uniform steel rod !ith a mass of 1$ &g. ind the moment of inertia of the barbell about the axis through the center *see figure).
Ans!er" 22 &gm$ #ar" 1
15) An extremely light rod 1.44 m long has a $.44-&g mass attached to one end and a .44-&g mass attached to the other. +he system rotates at a constant angular speed about a fixed axis perpendicular to the rod that passes through the rod 4.4 cm from the end !ith the .44-&g mass attached. +he &inetic energy of the system is measured to be 144.4 . *a) What is the moment of inertia of this system about the fixed axis *b) What is the angular speed *in revolutions per second) of this system Ans!er" *a) 1.$6 &gm$ *b) $.41 rev
17) A uniform solid sphere of mass M and radius R rotates !ith an angular speed ω about an axis through its center. A uniform solid cylinder of mass M , radius R, and length $ R rotates through an axis running through the central axis of the cylinder. What must be the angular speed of the cylinder so it !ill have the same rotational &inetic energy as the sphere A) $ω<6 B) ω C) 2ω<6 D) $ω< ) ω< Ans!er" D #ar" 1
18) While spinning do!n from 644.4 rpm to rest, a solid uniform fly!heel does (f the radius of the dis& is !hat is its mass A) 6.$ &g B) 2.2 &g C) 5.4 &g D) 5.8 &g Ans!er" A #ar" 64=
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of !or&.
1:) At any angular speed, a certain uniform solid sphere of diameter D has half as much rotational &inetic energy as a certain uniform thin-!alled hollo! sphere of the same diameter !hen both are spinning about an axis through their centers. (f the mass of the solid sphere is M , the mass of the hollo! sphere is A) <6 M. B) 6< M. C) 6<5 M. D) 5<6 M. ) $ M. Ans!er" D #ar" 1
$4) A futuristic design for a car is to have a large solid dis&-shaped fly!heel !ithin the car storing &inetic energy. +he uniform fly!heel has mass 74 &g !ith a radius of 4.644 m and can rotate up to Assuming all of this stored &inetic energy could be transferred to the linear velocity of the car, find the maximum attainable speed of the car. Ans!er" $25 m
$1) (n the figure, t!o bloc&s, of masses $.44 &g and .44 &g, are connected by a light string that passes over a frictionless pulley of moment of inertia 4.44244 &g m$ and radius 6.44 cm. +he coefficient of friction for the tabletop is 4.44. +he bloc&s are released from rest. Esing energy methods, find the speed of the upper bloc& ust as it has moved 4.544 m.
A) 1.$$ m
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$$) A 1.14-&g !rench is acting on a nut trying to turn it. +he length of the !rench lies directly to the east of the nut. A force 164.4 F acts on the !rench at a position 16.4 cm from the center of the nut in a direction 4.4@ north of east. What is the magnitude of the tor/ue about the center of the nut A) $$.6 Fm B) 11. Fm C) 1:.6 Fm D) $$64 Fm ) 1:2: Fm Ans!er" B #ar" 1
$) A :6 F force exerted at the end of a tor/ue !rench gives rise to a tor/ue of What is the angle *assumed to be less than :4@) bet!een the !rench handle and the direction of the applied force A) 18@ B) 12@ C) $$@ D) $6@ Ans!er" A #ar" 64=
$2) A light triangular plate OAB is in a hori%ontal plane. +hree forces, and
act on the plate, !hich is pivoted about a vertical axes through pointO. (n the
figure,
$ is perpendicular to OB. Consider the countercloc&!ise sense as positive. +he sum of the tor/ues about the vertical axis through point O, acting on the plate due to forces F 1, F $, and F , is closest to
A) 2.1 F m B) 6.2 F m C) -2.1 F m D) -6.2 F m ) %ero Ans!er" A #ar" 64=
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$6) A 7$.4-&g person pushes on a small door&nob !ith a force of 6.44 F perpendicular to the surface of the door. +he door&nob is located 4.844 m from axis of the frictionless hinges of the door. +he door begins to rotate !ith an angular acceleration of $.44 rad
$5) A tor/ue of 1$ F m is applied to a solid, uniform dis& of radius 4. 64 m, causing the dis& to accelerate at What is the mass of the dis& A) 17 &g B) 1 &g C) 8.6 &g D) 2. &g Ans!er" A #ar" 64=
$7) (n an experiment, a student brings up the rotational speed of a piece of laboratory apparatus to 4.4 rpm. Ghe then allo!s the apparatus to slo! do!n uniformly on its o!n, and counts $24 revolutions before the apparatus comes to a stop. +he moment of inertia of the apparatus is &no!n to be 4.4864 &gm$. What is the magnitude of the retarding tor/ue on the apparatus A) 4.42$6 Fm B) 4.16: Fm C) 4.4787 Fm D) 4.444$78 Fm ) 4.444415 Fm Ans!er" D #ar" 1
$8) A string is !rapped around a pulley !ith a radius of $.4 cm and no appreciable friction in its axle. +he pulley is initially not turning. A constant force of 64 F is applied to the string, !hich does not slip, causing the pulley to rotate and the string to un!ind. (f the string un!inds 1.$ m in 2.: s, !hat is the moment of inertia of the pulley A) 4.17 &gm$ B) 17 &gm$ C) 12 &gm$ D) 4.$4 &gm$ ) 4.417 &gm$ Ans!er" D #ar" 1
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$:) +he rotating systems sho!n in the figure differ only in that the t!o identical movable masses are positioned a distance r from the axis of rotation *left), or a distance r <$ from the axis of rotation *right). 9ou release the hanging bloc&s simultaneously from rest, and call t H the time ta&en by the bloc& on the left and t I the time ta&en by the bloc& on the right to reach the bottom, respectively. +he bar, pulley, and rope have negligible mass, the rope does not slip, and there is no friction in the axle of the pulley. Ender these conditions
A) t H 0
t I.
B) t H 0 t I. C) t H 0 t I. D) t H 0 $t I . ) t l 0 2t I. Ans!er" D #ar" 1
4) A solid uniform sphere of mass 1.86 &g and diameter 26.4 cm spins about an axle through its center. Gtarting !ith an angular velocity of $.24 rev
1) A very thin uniform rod, $.24 m long and of !eight 16 F, has a frictionless hinge at its lo!er end. (t starts out vertically from rest and falls, pivoting about the hinge. ust as it has rotated through an angle of 66.4@, !hat is the do!n!ard acceleration of the end farthest from the hinge A) 6.4$ m
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$) A very thin hori%ontal, $.44-m long, 6.44-&g uniform beam that lies along the east-!est direction is acted on by t!o forces. At the east end of the beam, a $44-F force pushes do!n!ard. At the !est end of the beam, a $44-F force pushes up!ard. What is the angular acceleration of the beam A) $24 rad
) (n the figure, a very light rope is !rapped around a !heel of radius R 0 $.4 meters and does not slip. +he !heel is mounted !ith frictionless bearings on an axle through its center. A bloc& of mass 12 &g is suspended from the end of the rope. When the system is released from rest it is observed that the bloc& descends 14 meters in $.4 seconds. What is the moment of inertia of the !heel
Ans!er" 62 &g m$ #ar" 1
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2) A !heel has a radius of 4.24 m and is mounted on frictionless bearings. A bloc& is suspended from a rope that is !ound on the !heel and attached to it *see figure). +he !heel is released from rest and the bloc& descends 1.6 m in $.44 s !ithout any slipping of the rope. +he tension in the rope during the descent of the bloc& is $4 F. What is the moment of inertia of the !heel
A) .6 &g m$ B) .7 &g m$ C) .: &g m$ D) 2.1 &g m$ ) 2. &g m$ Ans!er" #ar" 1
6) (n the figure, a mass of 1.77 &g is attached to a light string that is !rapped around a cylindrical spool of radius 14.4 cm and moment of inertia 2.44 &g m$. +he spool is suspended from the ceiling, and the mass is then released from rest a distance 6.74 m above the floor. >o! long does it ta&e to reach the floor
A) .:8 s B) .8 s C) 1.12 s D) 6.6: s ) 7.8: s Ans!er" A #ar" 64=
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5) or the apparatus sho!n in the figure, there is no slipping bet!een the cord and the surface of the pulley. +he bloc&s have mass of .4 &g and 6.7 &g, and the pulley has a radius of 4.1$ m and a mass of 14. &g. At the instant the 6.7 &g mass has fallen 1.6 m starting from rest, find the speed of each bloc&. *Assume the pulley is in the shape of a uniform solid dis& and has no friction in its axle.)
Ans!er" $.2 m
7) A uniform solid sphere of mass 1.6 &g and diameter 4.4 cm starts from rest and rolls !ithout slipping do!n a 6@ incline that is 7.4 m long. *a) Calculate the linear speed of the center of the sphere !hen it reaches the bottom of the incline. *b) Determine the angular speed of the sphere about its center at the bottom of the incline. *c) +hrough !hat angle *in radians) does this sphere turn as it rolls do!n the incline *d) Does the linear speed in *a) depend on the radius or mass of the sphere Does the angular speed in *b) depend on the radius or mass of the sphere Ans!er" *a) 7.6 m
8) A uniform solid 6.$6-&g cylinder is released from rest and rolls !ithout slipping do!n an inclined plane inclined at 18@ to the hori%ontal. >o! fast is it moving after it has rolled $.$ m do!n the plane A) 2. m
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:) A uniform solid cylinder of radius R and a thin uniform spherical shell of radius R both roll !ithout slipping. (f both obects have the same mass and the same &inetic energy, !hat is the ratio of the linear speed of the cylinder to the linear speed of the spherical shell A) <$ B)
<$
C) D) 2 < ) 2 < Ans!er" B #ar" 1
24) A uniform solid sphere is rolling !ithout slipping along a hori%ontal surface !ith a speed of 6.64 m
21) A solid, uniform sphere of mass $.4 &g and radius 1.7 m rolls from rest !ithout slipping do!n an inclined plane of height 7.4 m. What is the angular velocity of the sphere at the bottom of the inclined plane A) 6.8 rad
2$) A uniform solid dis& of radius 1.54 m and mass $.4 &g rolls !ithout slipping to the bottom of an inclined plane. (f the angular velocity of the dis& is at the bottom, !hat is the height of the inclined plane A) 6.51 m B) 2.$1 m C) 2.:2 m D) 5.7 m Ans!er" A #ar" 64=
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