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Vector Mechanics for Engineers: Statics and Dynamics

Ferdinand P. Beer, E. Russell Johnston, Jr., David F. Mazurek

Chapter 12

Kinetics of Particles: Newton's Second Law - all with Video Answers

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Chapter Questions

01:34

Problem 1

The acceleration due to gravity on Mars is $3.75 \mathrm{~m} / \mathrm{s}^{2}$. Knowing that the mass of a silver bar has been officially designated as $10 \mathrm{~kg},$ determine its weight in newtons on Mars.

Mukesh Devi
Mukesh Devi
Numerade Educator
06:50

Problem 2

The value of $g$ at any latitude $\phi$ may be obtained from the formula
$$
g=32.09\left(1+0.0053 \sin ^{2} \phi\right) \mathrm{ft} / \mathrm{s}^{2}$$which takes into account the effect of the rotation of the earth, as well as the fact that the earth is not truly spherical. Knowing that the weight of a silver bar has been officially designated as $5 \mathrm{lb},$ determine to four significant figures $(a)$ the mass in slugs, $(b)$ the weight in pounds at latitudes of $0^{\circ}, 30^{\circ},$ and $50^{\circ} .$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:22

Problem 3

A Global Positioning System (GPS) satellite is in a circular orbit $12,580 \mathrm{mi}$ above the surface of the earth and completes one orbit every 12 h. Knowing that the magnitude of the linear momentum of the satellite is $750 \times 10^{3} \mathrm{lb} \cdot \mathrm{s}$ and the radius of the earth is $3960 \mathrm{mi}$, determine ( $a$ ) the mass of the satellite, $(b)$ the weight of the satellite before it was launched from earth.

Andy Chen
Andy Chen
Numerade Educator
01:21

Problem 4

A spring scale $A$ and a lever scale $B$ having equal lever arms are fastened to the roof of an elevator, and identical packages are attached to the scales as shown. Knowing that when the elevator moves downward with an acceleration of $1 \mathrm{~m} / \mathrm{s}^{2}$ the spring scale indicates a load of $60 \mathrm{~N},$ determine $(a)$ the weight of the packages, $(b)$ the load indicated by the spring scale and the mass needed to balance the lever scale when the elevator moves upward with an acceleration of $1 \mathrm{~m} / \mathrm{s}^{2}$.

Salamat Ali
Salamat Ali
Numerade Educator
03:25

Problem 5

A loading car is at rest on a track forming an angle of $25^{\circ}$ with the vertical when a force is applied to the cable attached at $C$. The gross weight of the car and its load is $5500 \mathrm{lb},$ and it acts at point $G$. Knowing the tension in the cable connected at $C$ is 5000 lb, determine
(a) the acceleration of the car, (b) the distance the car moves in $20 \mathrm{~s}$,
(c) the time it takes for the car to return to its original position if the cable breaks after $20 \mathrm{~s}$

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
03:26

Problem 6

A $0.5-$ oz model rocket is launched vertically from rest at time $t=0$ with a constant thrust of 0.9 lb for $0.3 \mathrm{~s}$ and no thrust for $t>0.3 \mathrm{~s}$. Neglecting air resistance and the decrease in mass of the rocket, determine $(a)$ the maximum height $h$ reached by the rocket, $(b)$ the time required to reach this maximum height.

Christopher Dzorkpata
Christopher Dzorkpata
Numerade Educator
08:34

Problem 7

Determine the maximum theoretical speed that may be achieved over a distance of $60 \mathrm{~m}$ by a car starting from rest, knowing that the coefficient of static friction is 0.80 between the tires and the pavement and that 60 percent of the weight of the car is distributed over its front wheels and 40 percent over its rear wheels. Assume ( $a$ ) four-wheel drive, $(b)$ front-wheel drive, $(c)$ rear-wheel drive.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:17

Problem 8

A tugboat pulls a small barge through a harbor. The propeller thrust minus the drag produces a net thrust that varies linearly with speed. Knowing that the combined weight of the tug and barge is $3600 \mathrm{kN}$, determine ( $a$ ) the time required to increase the speed from an initial value $v_{1}=1.0 \mathrm{~m} / \mathrm{s}$ to a final value $v_{2}=2.5 \mathrm{~m} / \mathrm{s},(b)$ the distance traveled during this time interval.

Narayan Hari
Narayan Hari
Numerade Educator
04:44

Problem 9

If an automobile's braking distance from $108 \mathrm{~km} / \mathrm{h}$ is $75 \mathrm{~m}$ on level pavement, determine the automobile's braking distance from $108 \mathrm{~km} / \mathrm{h}$ when it is $(a)$ going up a $5^{\circ}$ incline, $(b)$ going down a 3-percent incline. Assume the braking force is independent of grade.

Supratim Pal
Supratim Pal
Numerade Educator
01:42

Problem 10

A 4 -kg package is released from rest at point $A$ and travels down the conveyor shown. Portions $A B$ and $C D$ are parallel to each other. Neglecting friction and any other energy loss, determine ( $a$ ) the acceleration of the package at $A,(b)$ the acceleration of the package during the horizontal portion of the track, $(c)$ the speed of the package at point $D .$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:59

Problem 11

The coefficients of friction between the load and the flatbed trailer shown are $\mu_{s}=0.40$ and $\mu_{k}=0.30 .$ Knowing that the speed of the rig is $72 \mathrm{~km} / \mathrm{h},$ determine the shortest distance in which the rig can be brought to a stop if the load is not to shift.

Salamat Ali
Salamat Ali
Numerade Educator
03:10

Problem 12

A light train made up of two cars is traveling at $90 \mathrm{~km} / \mathrm{h}$ when the brakes are applied to both cars. Knowing that car $A$ has a mass of $25 \mathrm{Mg}$ and car $B$ a mass of $20 \mathrm{Mg}$, and that the braking force is $30 \mathrm{kN}$ on each car, determine ( $a$ ) the distance traveled by the train before it comes to a stop, $(b)$ the force in the coupling between the cars while the train is slowing down.

Salamat Ali
Salamat Ali
Numerade Educator
08:21

Problem 13

The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and between the blocks and the incline, determine ( $a$ ) the acceleration of each block, (b) the tension in the cable.

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
06:32

Problem 14

The two blocks shown are originally at rest. Neglecting the masses of the pulleys and the effect of friction in the pulleys and knowing that the coefficients of friction between the blocks and the inclines are $\mu_{s}=0.25$ and $\mu_{i}=0.20,$ determine $(a)$ the acceleration of each block, (b) the tension in the cable.

Narayan Hari
Narayan Hari
Numerade Educator
14:58

Problem 15

Each of the systems shown is initially at rest. Neglecting axle friction and the masses of the pulleys, determine for each system ( $a$ ) the acceleration of block $A,(b)$ the velocity of block $A$ after it has moved through $10 \mathrm{ft},(c)$ the time required for block $A$ to reach a velocity of $20 \mathrm{ft} / \mathrm{s}$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:53

Problem 16

Boxes $A$ and $B$ are at rest on a conveyor belt that is initially at rest. The belt is suddenly started in an upward direction so that slipping occurs between the belt and the boxes. Knowing that the coefficients of kinetic friction between the belt and the boxes are $\left(\mu_{k}\right)_{A}=0.30$ and $\left(\mu_{i}\right)_{B}=0.32,$ determine the initial acceleration of each box.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:45

Problem 17

200-lb pallet $A$. Knowing the acceleration of the truck is $1 \mathrm{ft} / \mathrm{s}^{2},$ determine $(a)$ the horizontal force between the tires and the ground, $(b)$ the force between the boulder and the pallet.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
25:35

Problem 18

Block $A$ has a mass of $40 \mathrm{~kg},$ and block $B$ has a mass of $8 \mathrm{~kg} .$ The coefficients of friction between all surfaces of contact are $\mu_{s}=0.20$ and $\mu_{k}=0.15 .$ If $P=0,$ determine $(a)$ the acceleration of block $B$
(b) the tension in the cord.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
30:39

Problem 19

Block $A$ has a mass of $40 \mathrm{~kg},$ and block $B$ has a mass of $8 \mathrm{~kg} .$ The coefficients of friction between all surfaces of contact are $\mu_{s}=0.20$ and $\mu_{k}=0.15 .$ If $P=40 \mathrm{~N},$ determine $(a)$ the acceleration of block $B$
(b) the tension in the cord.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:15

Problem 20

The flat-bed trailer carries two $1500-\mathrm{kg}$ beams with the upper beam secured by a cable. The coefficients of static friction between the two beams and between the lower beam and the bed of the trailer are 0.25 and $0.30,$ respectively. Knowing that the load does not shift, determine ( $a$ ) the maximum acceleration of the trailer and the corresponding tension in the cable, (b) the maximum deceleration of the trailer.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:30

Problem 21

The position of the 10 -lb machine block $B$ is adjusted by moving the 5 -lb wedge $A$. Neglect friction between all surfaces of contact. Knowing that $\mathbf{P}=10$ lb, determine the $(a)$ acceleration of $B,(b)$ force between $A$ and $B$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
18:11

Problem 22

To unload a bound stack of plywood from a truck, the driver first tilts the bed of the truck and then accelerates from rest. Knowing that the coefficients of friction between the bottom sheet of plywood and the bed are $\mu_{s}=0.40$ and $\mu_{k}=0.30,$ determine $(a)$ the smallest acceleration of the truck which will cause the stack of plywood to slide, $(b)$ the acceleration of the truck which causes corner $A$ of the stack to reach the end of the bed in $0.9 \mathrm{~s}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:46

Problem 23

To transport a series of bundles of shingles $A$ to a roof, a contractor uses a motor-driven lift consisting of a horizontal platform $B C$ which rides on rails attached to the sides of a ladder. The lift starts from rest and initially moves with a constant acceleration $\mathbf{a}_{1}$ as shown. The lift then decelerates at a constant rate $\mathbf{a}_{2}$ and comes to rest at $D,$ near the top of the ladder. Knowing that the coefficient of static friction between a bundle of shingles and the horizontal platform is 0.30 , determine the largest allowable acceleration a $_{1}$ and the largest allowable deceleration a $_{2}$ if the bundle is not to slide on the platform.

Salamat Ali
Salamat Ali
Numerade Educator
02:41

Problem 24

An airplane has a mass of $25 \mathrm{Mg}$ and its engines develop a total thrust of $40 \mathrm{kN}$ during take-off. If the drag $\mathrm{D}$ exerted on the plane has a magnitude $D=2.25 v^{2},$ where $v$ is expressed in meters per second and $D$ in newtons, and if the plane becomes airborne at a speed of $240 \mathrm{~km} / \mathrm{h},$ determine $(a)$ the length of runway required for the plane to take off, $(b)$ the time required to take off.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
05:22

Problem 25

Determine the maximum theoretical speed that a $1225 \mathrm{~kg}$ automobile starting from rest can reach after traveling $400 \mathrm{~m}$ if air resistance is considered. Assume that the coefficient of static friction between the tires and the pavement is $0.70,$ that the automobile has front-wheel drive, that the front wheels support 62 percent of the automobile's weight, and that the aerodynamic drag $\mathbf{D}$ has a magnitude $D=0.575 v^{2},$ where $D$ and $v$ are expressed in newtons and $\mathrm{m} / \mathrm{s}$ respectively.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
05:05

Problem 26

A constant force $\mathbf{P}$ is applied to a piston and rod of total mass $m$ to make them move in a cylinder filled with oil. As the piston moves, the oil is forced through orifices in the piston and exerts on the piston a force of magnitude $k v$ in a direction opposite to the motion of the piston. Knowing that the piston starts from rest at $t=0$ and $x=0,$ show that the equation relating $\underline{x}, v,$ and $t,$ where $x$ is the distance traveled by the piston and $v$ is the speed of the piston, is linear in each of these variables.

Salamat Ali
Salamat Ali
Numerade Educator
07:51

Problem 27

A spring $A B$ of constant $k$ is attached to a support at $A$ and to a collar of mass $m$. The unstretched length of the spring is $l$. Knowing that the collar is released from rest at $x=x_{0}$ and neglecting friction between the collar and the horizontal rod, determine the magnitude of the velocity of the collar as it passes through point $C$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
13:23

Problem 28

Block $A$ has a mass of $10 \mathrm{~kg}$, and blocks $B$ and $C$ have masses of $5 \mathrm{~kg}$ each. Knowing that the blocks are initially at rest and that $B$ moves through $3 \mathrm{~m}$ in $2 \mathrm{~s}$, determine $(a)$ the magnitude of the force $\mathbf{P},(b)$ the tension in the cord $A D .$ Neglect the masses of the pulleys and axle friction.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
10:27

Problem 29

A 40-lb sliding panel is supported by rollers at $B$ and $C$. A 25 - Ib counterweight $A$ is attached to a cable as shown and, in cases $a$ and $c,$ is initially in contact with a vertical edge of the panel. Neglecting friction, determine in each case shown the acceleration of the panel and the tension in the cord immediately after the system is released from rest.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
05:59

Problem 30

An athlete pulls handle $A$ to the left with a constant force of $P=100 \mathrm{~N}$. Knowing that after the handle $A$ has been pulled $30 \mathrm{~cm}$ its velocity is $3 \mathrm{~m} / \mathrm{s},$ determine the mass of the weight stack $B$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
20:13

Problem 31

A 10-lb block $B$ rests as shown on a 20 -lb bracket $A$. The coefficients of friction are $\mu_{s}=0.30$ and $\mu_{k}=0.25$ between block $B$ and bracket $A$, and there is no friction in the pulley or between the bracket and the horizontal surface. (a) Determine the maximum weight of block $C$ if block $B$ is not to slide on bracket $A .(b)$ If the weight of block $C$ is 10 percent larger than the answer found in $a$, determine the accelerations of $A, B,$ and $C$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:15

Problem 32

Knowing that $\mu_{k}=0.30,$ determine the acceleration of each block when $m_{A}=m_{B}=m_{C}$

Salamat Ali
Salamat Ali
Numerade Educator
06:35

Problem 33

Knowing that $\mu_{k}=0.30,$ determine the acceleration of each block when $m_{A}=5 \mathrm{~kg}, m_{B}=30 \mathrm{~kg},$ and $m_{C}=15 \mathrm{~kg}$

Salamat Ali
Salamat Ali
Numerade Educator
05:29

Problem 34

The 30 - Ib block $B$ is supported by the 55 - Ib block $A$ and is attached to a cord to which a 50 -lb horizontal force is applied as shown. Neglecting friction, determine $(a)$ the acceleration of block $A$, (b) the acceleration of block $B$ relative to $A$.

Narayan Hari
Narayan Hari
Numerade Educator
15:26

Problem 35

Block $B$ of mass $10 \mathrm{~kg}$ rests as shown on the upper surface of a $22-\mathrm{kg}$ wedge $A .$ Knowing that the system is released from rest and neglecting friction, determine $(a)$ the acceleration of $B,(b)$ the velocity of $B$ relative to $A$ at $t=0.5 \mathrm{~s}$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:42

Problem 36

Knowing that the swings of an amusement park ride form an angle of $40^{\circ}$ with respect to the horizontal, determine $(a)$ the speed of rotation,
(b) the force in the cable for a swing and person weighing $250 \mathrm{lb}$.

Anand Jangid
Anand Jangid
Numerade Educator
02:22

Problem 37

During a hammer thrower's practice swings, the 7.1 -kg head $A$ of the hammer revolves at a constant speed in a horizontal circle as shown. Knowing that the speed of the hammer is $2.5 \mathrm{~m} / \mathrm{s}$ and $\theta=60^{\circ},$ determine $(a)$ the tension in wire $B C,(b)$ the radius of the circle, $\rho$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
05:08

Problem 38

Human centrifuges are often used to simulate different acceleration levels for pilots. When aerospace physiologists say that a pilot is pulling $9 \mathrm{~g}^{*} \mathrm{~s},$ they mean that the resultant normal force on the pilot from the bottom of the seat is nine times their weight. Knowing that the centrifuge starts from rest and has a constant angular acceleration of 1.5 RPM per second until the pilot is pulling $9 \mathrm{~g}^{\prime} \mathrm{s}$ and then continues with a constant angular velocity, determine $(a)$ how long it will take for the pilot to reach $9 \mathrm{~g}^{\prime} \mathrm{s}(b)$ the angle $\theta$ of the normal force once the pilot reaches $9 \mathrm{~g}^{\prime} \mathrm{s}$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:48

Problem 39

A single wire $A C B$ passes through a ring at $C$ attached to a sphere which revolves at a constant speed $v$ in the horizontal circle shown. Knowing that the tension is the same in both portions of the wire, determine the speed $v$

Salamat Ali
Salamat Ali
Numerade Educator
11:12

Problem 40

Two wires $A C$ and $B C$ are tied at $C$ to a sphere that revolves at a constant speed $v$ in the horizontal circle shown. Determine the range of the allowable values of $v$ if both wires are to remain taut and if the tension in either of the wires is not to exceed $60 \mathrm{~N}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:32

Problem 41

A 2 -lb sphere is at rest relative to a parabolic dish which rotates at a constant rate about a vertical axis. Neglecting friction and knowing that $r=3 \mathrm{ft},$ determine $(a)$ the velocity $v$ of the sphere, $(b)$ the magnitude of the normal force exerted by the sphere on the inclined surface of the dish.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
05:39

Problem 42

The $0.5-\mathrm{kg}$ flyballs of a centrifugal governor revolve at a constant speed $v$ in the horizontal circle of $150-\mathrm{mm}$ radius shown. Neglecting the mass of links $A B, B C, A D,$ and $D E,$ and requiring that the links support only tensile forces, determine the range of the allowable values of $v$ so that the magnitudes of the forces in the links do not exceed $75 \mathrm{~N}$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
05:21

Problem 43

As part of an outdoor display, a 5 -kg model $C$ of the earth is attached to wires $A C$ and $B C$ and revolves at a constant speed $v$ in the horizontal circle shown. Determine the range of the allowable values of $v$ if both wires are to remain taut and if the tension in either of the wires is not to exceed $116 \mathrm{~N}$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:29

Problem 44

A 130 -Ib wrecking ball $B$ is attached to a 45 -ft-long steel cable $A B$ and swings in the vertical arc shown. Determine the tension in the cable
(a) at the top $C$ of the swing, $(b)$ at the bottom $D$ of the swing, where the speed of $B$ is $13.2 \mathrm{ft} / \mathrm{s}$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:28

Problem 45

During a high-speed chase, a 2400 - -Ib sports car traveling at a speed of $100 \mathrm{mi} / \mathrm{h}$ just loses contact with the road as it reaches the crest $A$ of a hill. (a) Determine the radius of curvature $\rho$ of the vertical profile of the road at $A$. (b) Using the value of $\rho$ found in part $a$, determine the force exerted on a 160 -lb driver by the seat of his $3100-\mathrm{lb}$ car as the car, traveling at a constant speed of $50 \mathrm{mi} / \mathrm{h}$, passes through $A$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
View

Problem 46

An airline pilot climbs to a new flight level along the path shown. Knowing that the speed of the airplane decreases at a constant rate from $180 \mathrm{~m} / \mathrm{s}$ at point $A$ to $160 \mathrm{~m} / \mathrm{s}$ at point $C,$ determine the magnitude of the abrupt change in the force exerted on a $90-\mathrm{kg}$ passenger as the airplane passes point $B$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
04:26

Problem 47

The roller-coaster track shown is contained in a vertical plane. The portion of track between $A$ and $B$ is straight and horizontal, while the portions to the left of $A$ and to the right of $B$ have radii of curvature as indicated. A car is traveling at a speed of $72 \mathrm{~km} / \mathrm{h}$ when the brakes are suddenly applied, causing the wheels of the car to slide on the track $\left(\mu_{k}=0.20\right) .$ Determine the initial deceleration of the car if the brakes are applied as the car $(a)$ has almost reached $A,(b)$ is traveling between $A$ and $B,(c)$ has just passed $B$.

Salamat Ali
Salamat Ali
Numerade Educator
05:45

Problem 48

A spherical-cap governor is fixed to a vertical shaft that rotates with angular velocity $\omega$. When the string-supported clapper of mass $m$ touches the cap, a cutoff switch is operated electrically to reduce the speed of the shaft. Knowing that the radius of the clapper is small relative to the cap, determine the minimum angular speed at which the cutoff switch operates.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:48

Problem 49

A series of small packages, each with a mass of $0.5 \mathrm{~kg}$, are discharged from a conveyor belt as shown. Knowing that the coefficient of static friction between each package and the conveyor belt is $0.4,$ determine (a) the force exerted by the belt on the package just after it has passed point $A,(b)$ the angle $\theta$ defining the point $B$ where the packages first slip relative to the belt.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:58

Problem 50

A $55-\mathrm{kg}$ pilot flies a jet trainer in a half vertical loop of $1200-\mathrm{m}$ radius so that the speed of the trainer decreases at a constant rate. Knowing that the plane has a speed of $550 \mathrm{~km} / \mathrm{h}$ at point $A,$ and the pilot experiences weightlessness at point $C$ (i.e., the normal force from the seat bottom is zero), determine ( $a$ ) the deceleration of the plane, ( $b$ ) the force exerted on her by the seat of the trainer when the trainer is at point $B$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:46

Problem 51

A carnival ride is designed to allow the general public to experience high-acceleration motion. The ride rotates about point $O$ in a horizontal circle such that the rider has a speed $v_{0}$. The rider reclines on a platform $A$ which rides on rollers such that friction is negligible. A mechanical stop prevents the platform from rolling down the incline. Determine
(a) the speed $v_{0}$ at which the platform $A$ begins to roll upward, (b) the normal force experienced by an $80-\mathrm{kg}$ rider at this speed.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
10:57

Problem 52

A curve in a speed track has a radius of $1000 \mathrm{ft}$ and a rated speed of $120 \mathrm{mi} / \mathrm{h} .$ (See Sample Prob. 12.7 for the definition of rated speed.) Knowing that a racing car starts skidding on the curve when traveling at a speed of $180 \mathrm{mi} / \mathrm{h}$, determine $(a)$ the banking angle $\theta,(b)$ the coefficient of static friction between the tires and the track under the prevailing conditions, ( $c$ ) the minimum speed at which the same car could negotiate the curve.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:51

Problem 53

Tilting trains, such as the Acela Express that serves the Northeast Corridor in the Northeast United States, are designed to travel safely at high speeds on curved sections of track which were built for slower, conventional trains. As it enters a curve, each car is tilted by hydraulic actuators mounted on its trucks. The tilting feature of the cars also increases passenger comfort by eliminating or greatly reducing the side force $\mathbf{F}_{x}$ (parallel to the floor of the car) to which passengers feel subjected. For a train traveling at $100 \mathrm{mi} / \mathrm{h}$ on a curved section of track banked through an angle $\theta=6^{\circ}$ and with a rated speed of $60 \mathrm{mi} / \mathrm{h},$ determine $(a)$ the magnitude of the side force felt by a passenger of weight $W$ in a standard car with no tilt $(\phi=0),(b)$ the required angle of tilt $\phi$ if the passenger is to feel no side force. (See Sample Prob. 12.7 for the definition of rated speed.)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:12

Problem 54

Tests carried out with the tilting trains described in Prob. 12.53 revealed that passengers feel queasy when they see through the car windows that the train is rounding a curve at high speed, yet do not feel any side force. Designers, therefore, prefer to reduce, but not eliminate that force. For the train of Prob. 12.53 , determine the required angle of tilt $\phi$ if passengers are to feel side forces equal to 10 percent of their weights.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:24

Problem 55

A 3 -kg block is at rest relative to a parabolic dish which rotates at a constant rate about a vertical axis. Knowing that the coefficient of static friction is 0.5 and that $r=2 \mathrm{~m},$ determine the maximum allow able velocity $v$ of the block.

Salamat Ali
Salamat Ali
Numerade Educator
02:37

Problem 56

A polisher is started so that the fleece along the circumference undergoes a constant tangential acceleration of $4 \mathrm{~m} / \mathrm{s}^{2} .$ Three seconds after it is started, small tufts of fleece from along the circumference of the 225-mm-diameter polishing pad are observed to fly free of the pad. At this instant, determine $(a)$ the speed $v$ of a tuft as it leaves the pad, (b) the magnitude of the force required to free a tuft if the average mass of a tuft is $1.6 \mathrm{mg}$.

Salamat Ali
Salamat Ali
Numerade Educator
02:14

Problem 57

A turntable $A$ is built into a stage for use in a theatrical production. It is observed during a rehearsal that a trunk $B$ starts to slide on the turntable $10 \mathrm{~s}$ after the turntable begins to rotate. Knowing that the trunk undergoes a constant tangential acceleration of $0.24 \mathrm{~m} / \mathrm{s}^{2},$ determine the coefficient of static friction between the trunk and the turntable.

Salamat Ali
Salamat Ali
Numerade Educator
01:57

Problem 58

The carnival ride from Prob. 12.51 is modified so that the $80-\mathrm{kg}$ riders can move up and down the inclined wall as the speed of the ride increases. Assuming that the friction between the wall and the carriage is negligible, determine the position $h$ of the rider if the speed $v_{0}=13 \mathrm{~m} / \mathrm{s}$.

Salamat Ali
Salamat Ali
Numerade Educator
14:16

Problem 59

The carnival ride from Prob 12.51 is modified so that the $80-\mathrm{kg}$ riders can move up and down the inclined wall as the speed of the ride increases. Knowing that the coefficient of static friction between the wall and the platform is $0.2,$ determine the range of values of the constant speed $v_{0}$ for which the platform will remain at $h=1.5 \mathrm{~m}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
14:49

Problem 60

A small $8-0 z$ collar $D$ can slide on portion $A B$ of a rod which is bent as shown. Knowing that the rod rotates about the vertical $A C$ at a constant rate and that $\alpha=40^{\circ}$ and $r=24$ in., determine the range of values of the speed $v$ for which the collar will not slide on the rod if the coefficient of static friction between the rod and the collar is 0.35 .

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:18

Problem 61

A small block $B$ fits inside a slot cut in arm $O A$ that rotates in a vertical plane at a constant rate. The block remains in contact with the end of the slot closest to $A$ and its speed is $1.4 \mathrm{~m} / \mathrm{s}$ for $0 \leq \theta \leq 150^{\circ} .$ Knowing that the block begins to slide when $\theta=150^{\circ},$ determine the coefficient of static friction between the block and the slot.

Salamat Ali
Salamat Ali
Numerade Educator
09:47

Problem 62

The parallel-link mechanism $A B C D$ is used to transport a component $I$ between manufacturing processes at stations $E, F,$ and $G$ by picking it up at a station when $\theta=0$ and depositing it at the next station when $\theta=180^{\circ} .$ Knowing that member $B C$ remains horizontal throughout its motion and that links $A B$ and $C D$ rotate at a constant rate in a vertical plane in such a way that $v_{B}=2.2 \mathrm{ft} / \mathrm{s},$ determine $(a)$ the minimum value of the coefficient of static friction between the component and $B C$ if the component is not to slide on $B C$ while being transferred,
(b) the values of $\theta$ for which sliding is impending.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
14:30

Problem 63

Knowing that the coefficients of friction between the component $I$ and member $B C$ of the mechanism of Prob. 12.62 are $\mu_{s}=0.35$ and $\mu_{k}=0.25,$ determine $(a)$ the maximum allowable constant speed $v_{b}$ if the component is not to slide on $B C$ while being transferred, ( $b$ ) the values of $\theta$ for which sliding is impending.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:15

Problem 64

A small $250-g$ collar $C$ can slide on a semicircular rod which is made to rotate about the vertical $A B$ at a constant rate of $7.5 \mathrm{rad} / \mathrm{s}$. Determine the three values of $\theta$ for which the collar will not slide on the rod, assuming no friction between the collar and the rod.

Salamat Ali
Salamat Ali
Numerade Educator
14:49

Problem 65

A small $250-\mathrm{g}$ collar $C$ can slide on a semicircular rod which is made to rotate about the vertical $A B$ at a constant rate of $7.5 \mathrm{rad} / \mathrm{s}$. Knowing that the coefficients of friction are $\mu_{s}=0.25$ and $\mu_{k}=0.20,$ indicate whether the collar will slide on the rod if it is released in the position corresponding to $(a) \theta=75^{\circ},(b) \theta=40^{\circ} .$ Also, determine the magnitude and direction of the friction force exerted on the collar immediately after release.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:50

Problem 66

An advanced spatial disorientation trainer allows the cab to rotate around multiple axes, as well as to extend inward and outward. It can be used to simulate driving, fixed-wing aircraft flying, and helicopter maneuvering. In one training scenario, the trainer rotates and translates in the horizontal plane, where the location of the pilot is defined by the relationships $r=10+2 \cos (4 t)$ and $\theta=0.1\left(2 t^{2}-1\right),$ where $r, \theta,$ and $t$ are expressed in feet, radians, and seconds, respectively. Knowing that the pilot has a weight of 175 lbs, (a) determine the magnitude of the resulting force acting on the pilot at $t=5 \mathrm{~s},(b)$ plot the magnitudes of the radial and transverse components of the force exerted on the pilot from 0 to 10 seconds.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
07:57

Problem 67

An advanced spatial disorientation trainer is programmed to only rotate and translate in the horizontal plane. The pilot's location is defined by the relationships $r=8\left(1-e^{-t}\right)$ and $\theta=2 / \pi\left(\sin \frac{\pi}{2} t\right),$ where $r, \theta,$ and $t$ are expressed in feet, radians, and seconds, respectively. Determine the radial and transverse components of the force exerted on the 175 -lb pilot at $t=3 \mathrm{~s}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:47

Problem 68

The $3-\mathrm{kg}$ collar $B$ slides on the frictionless arm $A A^{\prime} .$ The arm is attached to drum $D$ and rotates about $O$ in a horizontal plane at the rate $\dot{\theta}=0.75 t$, where $\dot{\theta}$ and $t$ are expressed in $\mathrm{rad} / \mathrm{s}$ and seconds, respectively. As the arm-drum assembly rotates, a mechanism within the drum releases cord so that the collar moves outward from $O$ with a constant speed of $0.5 \mathrm{~m} / \mathrm{s}$. Knowing that at $t=0, r=0$, determine the time at which the tension in the cord is equal to the magnitude of the horizontal force exerted on $B$ by arm $A A^{\prime}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:20

Problem 69

A 0.5-kg block $B$ slides without friction inside a slot cut in arm $O A$ that rotates in a vertical plane. The rod has a constant angular acceleration $\ddot{\theta}=10 \mathrm{rad} / \mathrm{s}^{2} .$ Knowing that when $\theta=45^{\circ}$ and $r=0.8 \mathrm{~m}$ the velocity of the block is zero, determine at this instant, ( $a$ ) the force exerted on the block by the arm, (b) the relative acceleration of the block with respect to the arm.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
11:52

Problem 70

Pin $B$ weighs 4 oz and is free to slide in a horizontal plane along the rotating arm $O C$ and along the fixed circular slot $D E$ of radius $b=20$ in. Neglecting friction and assuming that $\theta=15 \mathrm{rad} / \mathrm{s}$ and $\bar{\theta}=250 \mathrm{rad} / \mathrm{s}^{2}$ for the position $\theta=20^{\circ},$ determine for that position
(a) the radial and transverse components of the resultant force exerted on $\operatorname{pin} B,(b)$ the forces $\mathbf{P}$ and $\mathbf{Q}$ exerted on $\operatorname{pin} B,$ respectively, by rod $O C$ and the wall of slot $D E .$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:24

Problem 71

The parasailing system shown uses a winch to let rope out at a constant rate so that the 70 -kg rider moves away from the boat, which is traveling with a constant velocity. At the instant shown, the rope has a length of $30 \mathrm{~m}$, it is increasing in length at a constant $1 \mathrm{~m} / \mathrm{s}$, the angle is increasing at a rate of $0.05 \mathrm{rad} / \mathrm{s},$ and $\dot{\theta}$ is $-0.01 \mathrm{rad} / \mathrm{s}^{2} .$ Knowing that when the rope makes a $30^{\circ}$ angle with respect to the water, the tension in the rope is $10 \mathrm{kN}$, determine the magnitude and direction of the force of the parasail on the parasailor.

Mayukh Banik
Mayukh Banik
Numerade Educator
07:21

Problem 72

A $700-\mathrm{kg}$ horse $A$ lifts a $50-\mathrm{kg}$ hay bale $B$ as shown. At the instant when $r=8 \mathrm{~m}$ and $\theta=60^{\circ}$, the velocity and acceleration of the horse are $2 \mathrm{~m} / \mathrm{s}$ to the right and $0.5 \mathrm{~m} / \mathrm{s}^{2}$ to the left, respectively. Neglecting the mass of the pulley, determine at that instant ( $a$ ) the tension in the cable, (b) the average horizontal force between the ground and the horse's feet.

Donald Albin
Donald Albin
Numerade Educator
12:28

Problem 73

Slider $C$ has a weight of $0.5 \mathrm{lb}$ and may move in a slot cut in $\operatorname{arm} A B$, which rotates at the constant rate $\dot{\theta}_{0}=10 \mathrm{rad} / \mathrm{s}$ in a horizontal plane. The slider is attached to a spring of constant $k=2.5$ lb/ft, which is unstretched when $r=0 .$ Knowing that the slider is released from rest with no radial velocity in the position $r=18$ in. and neglecting friction, determine for the position $r=12$ in. $(a)$ the radial and transverse components of the velocity of the slider, $(b)$ the radial and transverse components of its acceleration, ( $c$ ) the horizontal force exerted on the slider by $\operatorname{arm} A B$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:08

Problem 74

A particle of mass $m$ is projected from point $A$ with an initial velocity $v_{0}$ perpendicular to line $O \mathrm{~A}$ and moves under a central force $\mathrm{F}$ dinected away from the center of force $0 .$ Knowing that the particle follows a path defined by the equation $r=r_{0} / \sqrt{\cos 2 \theta}$ and using $\mathrm{Eq} \cdot(12.25)$ express the radial and transverse components of the velocity $\mathbf{v}$ of the purticle as functions of $a$

Keshav Singh
Keshav Singh
Numerade Educator
14:24

Problem 75

For the particle of Prob. 12.74 , show (a) that the velocity of the particle and the central force $\mathrm{F}$ are proportional to the distance $r$ from the particle to the center of force $O,(b)$ that the radius of curvature of the path is proportional to $r$.

Keshav Singh
Keshav Singh
Numerade Educator
03:58

Problem 76

A particle of mass $m$ is projected from point $A$ with an initial velocity $v_{0}$ perpendicular to line $O A$ and moves under a central force $F$ along a semicircular path of diameter $O A$. Observing that $r=r_{0}$ cos $\theta$ and using Eq. $(12.25),$ show that the speed of the particle is $v=v_{j} / \cos ^{2} \theta$

Keshav Singh
Keshav Singh
Numerade Educator
08:28

Problem 77

For the particle of Prob. 12.76 , determine the tangential component $F_{i}$ of the central force $\mathrm{F}$ along the tangent to the path of the particle for
$(a) \theta=0,(b) \theta=45^{\circ}$

Keshav Singh
Keshav Singh
Numerade Educator
05:55

Problem 78

Determine the mass of the earth knowing that the mean radius of the moon's orbit about the earth is $238.910 \mathrm{mi}$ and that the moon requires 27.32 days to complete one full revolution about the earth.

Keshav Singh
Keshav Singh
Numerade Educator
07:29

Problem 79

Show that the radius $r$ of the moon's orbit ean be determined from the radius $R$ of the earth, the acceleration of gravity $g$ at the surfice of the earth, and the time r required for the moon to complete one full revolution about the earth. Compute $r$ knowing that $\tau=27.3$ days, giving the answer in both $\mathrm{Sl}$ and $\mathrm{U}$. S. customary units.

Keshav Singh
Keshav Singh
Numerade Educator
07:57

Problem 80

Communication satellites are placed in a geosynchronous orbit - that is, in a circular orbit such that they complete one full revolution about the earth in one sidereal day $(23.934 \mathrm{~h}),$ and thus appear stationary with respect to the ground Determine (a) the altitude of these satellites above the surface of the earth, (b) the velocity with which they describe their orbit. Give the answers in both $\mathrm{SI}$ and $\mathrm{U}$. S. customary units.

Keshav Singh
Keshav Singh
Numerade Educator
05:52

Problem 81

Show that the radius $r$ of the orbit of a moon of a given planet can be determined from the radius $R$ of the planet, the acceleration of gravity at the surface of the planet, and the time r required by the moon to complete one full revolution about the planet. Determine the acceleration of gravity at the surface of the planet Jupiter knowing that $R=71492 \mathrm{~km}$ and that $r=3.551$ days and $r=670.9 \times 10^{3} \mathrm{~km}$ for its moon Furopa.

Keshav Singh
Keshav Singh
Numerade Educator
05:31

Problem 82

The orbit of the planet Venus is nearly circular with an orbital velocity of $126.5 \times 10^{3}$ km/h. Knowing that the mean distance from the center of the sun to the center of Venus is $108 \times 10^{6} \mathrm{~km}$ and that the radius of the sun is $695.5 \times 10^{3} \mathrm{~km}$, determine $(a)$ the mass of the $\mathrm{sun},$ (b) the acceleration of gravity at the surface of the sun.

Keshav Singh
Keshav Singh
Numerade Educator
08:32

Problem 83

A satellite is placed into a circular orbit about the planet Saturn at an altitude of $2100 \mathrm{ml}$. The satellite describes its orbit with a velocity of $54.7 \times 10^{3} \mathrm{mi} / \mathrm{h} .$ Knowing that the radius of the orbit about Saturn and the periodic time of Atlas, one of Saturn's moons, are $85.54 \times 10^{3} \mathrm{mi}$ and 0.6017 days, respectively, determine $(a)$ the radius of Saturn,
(b) the mass of Saturn. (The periodic time of a satellite is the time it requires to complete one full revolution about the planet.

Keshav Singh
Keshav Singh
Numerade Educator
07:22

Problem 84

The periodic time (see Prob. 12.83 ) of an earth satellite in a circular. polar orbit is 120 minutes, Determine $(a)$ the altitude $h$ of the satellite, (b) the time during which the satellite is above the horizon for an observer located at the north pole.

Keshav Singh
Keshav Singh
Numerade Educator
10:56

Problem 85

A 500 -kg spacecraft fint is placed into a circular orbit about the earth at an altitude of $4500 \mathrm{~km}$ and then is transferred to a circular orbit about the moon. Knowing that the mass of the moon is 0.01230 times the mass of the earth and that the radius of the moon is $1737 \mathrm{~km}$, determine (a) the gravitational force exerted on the spacecraft as it was orbiting the earth, (b) the required radius of the orbit of the spacecraft about the $\mathrm{m} \% \mathrm{n}$ if the periodic times (see Prob. 12.83 ) of the two orbits are to be equal, $(c)$ the acceleration of gravity at the surface of the moon.

Keshav Singh
Keshav Singh
Numerade Educator
07:23

Problem 86

A space vehicle is in a circular orbit of 2200 - km radius around the moon. To transfer it to a smaller circular orbit of 2080 -km radius, the vehicle is first placed on an elliptic path $A B$ by reducing its speed by $26,3 \mathrm{~m} / \mathrm{s}$ as it passes through $\mathrm{A}$. Knowing that the mass of the moon is $73.49 \times 10^{21} \mathrm{~kg}$, determine $(a)$ the speed of the vehicle as it approaches $B$ on the elliptic path, $(b)$ the amount by which its speed should be reduced as it approaches $B$ to insert it into the smaller circular orbit.

Keshav Singh
Keshav Singh
Numerade Educator
10:03

Problem 87

As a first approximation to the analysis of a space flight from the earth to the planet Mins, assume the orbits of the earth and Mars are circular and co-planar. The mean distances from the sun to the earth and to Mars are $149.6 \times 10^{6} \mathrm{~km}$ and $227.8 \times 10^{6} \mathrm{~km}$, respectively. To place the spacecraft into an elliptical transfer orbit at point $A$, its speed is increased over a short interval of time to $v_{A}$ which is $2.94 \mathrm{~km} / \mathrm{s}$ faster than the earth's orbital speed. When the spacecraft reaches point $B$ on the elliptical transfer orbit, its speed $v_{n}$ is increased to the orbital speed of Mars, Knowing that the mass of the sun is $332.8 \times 10^{4}$ times the mass of the earth, determine the increase in speed required at $B$.

Keshav Singh
Keshav Singh
Numerade Educator
02:59

Problem 88

To place a communications satellite into a geosynchronous orbit (see Prob. 12.80 ) at an altitude of $22,240 \mathrm{mi}$ above the surface of the earth, the satellite first is released from a space shuttle, which is in a circular orbit at an altitude of $185 \mathrm{mi}$, and then is propelled by an upper stage booster to its final altitude. As the satellite passes through $A,$ the booster's motor is fired to insert the satellite into an elliptical transfer orbit. The booster is again fired at $B$ to insert the satellite into a geosynchronous orbit. Knowing that the second firing increases the speed of the satellite by $4810 \mathrm{ft} / \mathrm{s}$, determine $(a)$ the speed of the satellite as it approaches $B$ on the elliptic transfer orbit, (b) the increase in speed resulting from the first firing at $A .$

Anand Jangid
Anand Jangid
Numerade Educator
07:28

Problem 89

A space vehicle is in a circular orbit with a $1400-\mathrm{mi}$ radius around the moon. To transfer to a smaller orbit with a 1300 -mi radius, the vehicle is first placed in an elliptic path $A B$ by reducing its speed by $86 \mathrm{f} / \mathrm{s}$ as it passes through $A .$ Knowing that the mass of the moon is $5.03 \times 10^{21} \mathrm{lb} \cdot \mathrm{s}^{2} / \mathrm{ft},$ determine $(a)$ the speed of the vehicle as it approaches $B$ on the elliptic path, $(b)$ the amount by which its speed should be reduced as it approaches $B$ to insert it into the smaller circular orbit.

Keshav Singh
Keshav Singh
Numerade Educator
06:20

Problem 90

A 1-kg collar can slide on a horizontal rod that is free to rotate about a vertical shaft. The collar is initially held at $A$ by a cord attached to the shaft. A spring of constant $30 \mathrm{~N} / \mathrm{m}$ is attached to the collar and to the shaft and is undeformed when the collar is at $A$. As the rod rotates at the rate $\dot{\theta}=16 \mathrm{rad} / \mathrm{s},$ the cord is cut and the collar moves out along the rod. Neglecting friction and the mass of the rod, determine ( $a$ ) the radial and transverse components of the acceleration of the collar at $A$,
(b) the acceleration of the collar relative to the rod at $A,(c)$ the transverse component of the velocity of the collar at $B$.

Keshav Singh
Keshav Singh
Numerade Educator
01:37

Problem 91

A 1-lb ball $A$ and a 2 -lb ball $B$ are mounted on a horizontal rod that rotates freely about a vertical shaft. The balls are held in the positions shown by pins. The pin holding $B$ is suddenly removed and the ball moves to position $C$ as the rod rotates. Neglecting friction and the mass of the rod and knowing that the initial speed of $A$ is $v_{A}=8 \mathrm{ft} / \mathrm{s}$ determine ( $a$ ) the radial and transverse components of the acceleration of ball $B$ immediately after the pin is removed, $(b)$ the acceleration of ball $B$ relative to the rod at that instant, ( $c$ ) the speed of ball $A$ after ball $B$ has reached the stop at $C$.

Penny Riley
Penny Riley
Numerade Educator
10:05

Problem 92

Two $2.6-\mathrm{lb}$ collars $A$ and $B$ can slide without friction on a frame, consisting of the horizontal rod $O E$ and the vertical rod $C D,$ which is free to rotate about $C D$. The two collars are connected by a cord running over a pulley that is attached to the frame at $O,$ and a stop prevents collar $B$ from moving. The frame is rotating at the rate $\dot{\theta}=12$ rad/s and $r=0.6 \mathrm{ft}$ when the stop is removed, allowing collar $A$ to move out along rod $O E$. Neglecting friction and the mass of the frame, determine, for the position $r=1.2 \mathrm{ft},(a)$ the transverse component of the velocity of collar $A,(b)$ the tension in the cord and the acceleration of collar $A$ relative to the rod $O E$.

Keshav Singh
Keshav Singh
Numerade Educator
08:22

Problem 93

A small ball swings in a horizontal circle at the end of a cord of length $l_{1}$ which forms an angle $\theta_{1}$ with the vertical. The cord is then slowly drawn through the support at $O$ until the length of the free end is $l_{2}$. (a) Derive a relation among $l_{1}, l_{2}, \theta_{1,}$ and $\theta_{2} .(b)$ If the ball is set in motion so that initially $I_{1}=0.8 \mathrm{~m}$ and $\theta_{1}=35^{\circ},$ determine the angle $\theta_{2}$ when $l_{2}=0.6 \mathrm{~m}$.

Keshav Singh
Keshav Singh
Numerade Educator
04:13

Problem 94

A particle of mass $m$ is projected from point $A$ with an initial velocity $v_{0}$ perpendicular to $O A$ and moves under a central force $\mathbf{F}$ along an elliptic path defined by the equation $r=r_{0} /(2-\cos \theta) .$ Using Eq. ( 12.35 ), show that $\mathbf{F}$ is inversely proportional to the square of the distance $r$ from the particle to the center of force $O$.

Deepak Kohli
Deepak Kohli
Numerade Educator
03:18

Problem 95

A particle of mass $m$ describes the logarithmic spiral $r=r_{0} e^{b e}$ under a central force $\mathbf{F}$ directed toward the center of force $O$. Using Eq. (12.35), show that $\mathbf{F}$ is inversely proportional to the cube of the distance $r$ from the particle to $O$.

Deepak Kohli
Deepak Kohli
Numerade Educator
02:32

Problem 96

A particle with a mass $m$ describes the path defined by the equation $r=r_{0} /(6 \cos \theta-5)$ under a central force $\mathbf{F}$ directed away from the center of force $O$. Using Eq. (12.35), show that $\mathbf{F}$ is inversely proportional to the square of the distance $r$ from the particle to $O$.

Deepak Kohli
Deepak Kohli
Numerade Educator
09:05

Problem 97

A particle of mass $m$ describes the parabola $y=x^{2} / 4 r_{0}$ under a central force $\mathbf{F}$ directed toward the center of force $C$. Using Eq. (12.35) and Eq. $\left(12.37^{\prime}\right)$ with $\varepsilon=1,$ show that $\mathbf{F}$ is inversely proportional to the square of the distance $r$ from the particle to the center of force and that the angular momentum per unit mass $h=\sqrt{2 G M r_{0}}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:34

Problem 98

It was observed that during its second flyby of the earth, the Galileo spacecraft had a velocity of $14.1 \mathrm{~km} / \mathrm{s}$ as it reached its minimum altitude of $303 \mathrm{~km}$ above the surface of the earth. Determine the eccentricity of the trajectory of the spacecraft during this portion of its flight.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:05

Problem 99

It was observed that during the Galileo spacecraft's first flyby of the earth, its minimum altitude was $600 \mathrm{mi}$ above the surface of the earth. A ssuming that the trajectory of the spacecraft was parabolic, determine the maximum velocity of Galileo during its first flyby of the earth.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:17

Problem 100

As a space probe approaching the planet Venus on a parabolic trajectory reaches point $A$ closest to the planet, its velocity is decreased to insert it into a circular orbit. Knowing that the mass and the radius of Venus are $4.87 \times 10^{24} \mathrm{~kg}$ and $6052 \mathrm{~km}$, respectively, determine ( $a$ ) the velocity of the probe as it approaches $A,(b)$ the decrease in velocity required to insert it into the circular orbit.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:27

Problem 101

It was observed that as the Voyager I spacecraft reached the point of its. trajectory closest to the planet Saturn, it was at a distance of $185 \times 10^{3} \mathrm{~km}$ from the center of the planet and had a velocity of $21.0 \mathrm{~km} / \mathrm{s}$. Knowing that Tethys, one of Saturn's moons, describes a circular orbit of radius $295 \times 10^{3} \mathrm{~km}$ at a speed of $11.35 \mathrm{~km} / \mathrm{s}$, determine the eccentricity of the trajectory of Voyager I on its approach to Saturn.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:50

Problem 102

A satellite describes an elliptic orbit about a planet of mass $M$. Denoting by $r_{0}$ and $r_{1},$ respectively, the minimum and maximum values of the distance $r$ from the satellite to the center of the planet, derive the relation
$$\frac{1}{r_{0}}+\frac{1}{r_{1}}=\frac{2 G M}{h^{2}}$$
where $h$ is the angular momentum per unit mass of the satellite.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:17

Problem 103

A space probe is describing a circular orbit about a planet of radius $R$ The altitude of the probe above the surface of the planet is $\alpha R$ and its speed is $v_{0}$ - To place the probe in an elliptic orbit which will bring it closer to the planet, its speed is reduced from $v_{0}$ to $\beta v_{0}$, where $\beta<1$, by firing its engine for a short interval of time. Determine the smallest permissible value of $\beta$ if the probe is not to crash on the surface of the planet.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
14:23

Problem 104

A satellite describes a circular orbit at an altitude of $19110 \mathrm{~km}$ above the surface of the earth. Determine (a) the increase in speed required at point $A$ for the satellite to achieve the escape velocity and enter a parabolic orbit, (b) the decrease in speed required at point $A$ for the satellite to enter an elliptic orbit with a minimum altitude of $6370 \mathrm{~km}$,
(c) the eccentricity $\varepsilon$ of the elliptic orbit.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
23:23

Problem 105

A space probe is to be placed in a circular orbit of $5600-\mathrm{mi}$ radius about the planet Venus in a specified plane. As the probe reaches $A$, the point of its original trajectory closest to Venus, it is inserted in a first elliptic transfer orbit by reducing its speed by $\Delta v_{A}$. This orbit brings it to point $B$ with a much reduced velocity. There the probe is inserted in a second transfer orbit located in the specified plane by changing the direction of its velocity and further reducing its speed by $\Delta v_{B}$. Finally, as the probe reaches point $C$, it is inserted in the desired circular orbit by reducing its speed by $\Delta v_{C}$. Knowing that the mass of Venus is 0.82 times the mass of the earth, that $r_{A}=9.3 \times 10^{3} \mathrm{mi}$ and $r_{B}=190 \times 10^{3} \mathrm{mi},$ and that the probe approaches $A$ on a parabolic trajectory, determine by how much the velocity of the probe should be reduced $(a)$ at $A,(b)$ at $B,(c)$ at $C$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
17:50

Problem 106

For the space probe of Prob. $12.105,$ it is known that $r_{A}=9.3 \times 10^{3} \mathrm{mi}$ and that the velocity of the probe is reduced to $20,000 \mathrm{ft} / \mathrm{s}$ as it passes through $A .$ Determine $(a)$ the distance from the center of Venus to point $B,(b)$ the amounts by which the velocity of the probe should be reduced at $B$ and $C$, respectively.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
18:33

Problem 107

As it describes an elliptic orbit about the sun, a spacecraft reaches a maximum distance of $202 \times 10^{6} \mathrm{mi}$ from the center of the sun at point $A$ (called the aphelion) and a minimum distance of $92 \times 10^{6} \mathrm{mi}$ at point $B$ (called the perihelion). To place the spacecraft in a smaller elliptic orbit with aphelion at $A^{\prime}$ and perihelion at $B^{\prime},$ where $A^{\prime}$ and $B^{\prime}$ are located $164.5 \times 10^{6} \mathrm{mi}$ and $85.5 \times 10^{6} \mathrm{mi},$ respectively, from the center of the sun, the speed of the spacecraft is first reduced as it passes through $A$ and then is further reduced as it passes through $B^{\prime}$. Knowing that the mass of the sun is $332.8 \times 10^{3}$ times the mass of the earth, determine $(a)$ the speed of the spacecraft at $A,(b)$ the amounts by which the speed of the spacecraft should be reduced at $A$ and $B^{\prime}$ to insert it into the desired elliptic orbit.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:06

Problem 108

Halley's comet travels in an elongated elliptic orbit for which the minimum distance from the sun is approximately $\downarrow r_{E},$ where $r_{E}=150 \times 10^{6} \mathrm{~km}$ is the mean distance from the sun to the earth. Knowing that the periodic time of Halley's comet is about 76 years, determine the maximum distance from the sun reached by the comet.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:13

Problem 109

Based on observations made during the 1996 sighting of comet Hyakutake, it was concluded that the trajectory of the comet is a highly elongated ellipse for which the eccentricity is approximately $\varepsilon=0.999887$. Knowing that for the 1996 sighting the minimum distance between the comet and the sun was $0.230 R_{E},$ where $R_{E}$ is the mean distance from the sun to the earth, determine the periodic time of the comet.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
10:42

Problem 110

A space probe is to be placed in a circular orbit of radius $4000 \mathrm{~km}$ about the planet Mars. As the probe reaches $A$, the point of its original trajectory closest to Mars, it is inserted into a first elliptic transfer orbit by reducing its speed. This orbit brings it to point $B$ with a much-reduced velocity. There the probe is inserted into a second transfer orbit by further reducing its speed. Knowing that the mass of Mars is 0.1074 times the mass of the earth, that $r_{A}=9000 \mathrm{~km}$ and $r_{B}=180000 \mathrm{~km},$ and that the probe approaches $A$ on a parabolic trajectory, determine the time needed for the space probe to travel from $A$ to $B$ on its first transfer orbit.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
15:33

Problem 111

A spacecraft and a satellite are at diametrically opposite positions in the same circular orbit of altitude $500 \mathrm{~km}$ above the earth. As it passes through point $A,$ the spacecraft fires its engine for a short interval of time to increase its speed and enter an elliptic orbit. Knowing that the spacecraft returns to $A$ at the same time the satellite reaches $A$ after completing one and a half orbits, determine ( $a$ ) the increase in speed required, (b) the periodic time for the elliptic orbit.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
10:00

Problem 112

The Clementine spacecraft described an elliptic orbit of minimum altitude $h_{A}=400 \mathrm{~km}$ and maximum altitude $h_{B}=2940 \mathrm{~km}$ above the surface of the moon. Knowing that the radius of the moon is $1737 \mathrm{~km}$ and that the mass of the moon is 0.01230 times the mass of the earth, determine the periodic time of the spacecraft.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
10:50

Problem 113

Determine the time needed for the space probe of Prob. 12.100 to travel from $B$ to $C$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:58

Problem 114

A space probe is describing a circular orbit of radius $n R$ with a velocity $v_{0}$ about a planet of radius $R$ and center $O$. As the probe passes through point $A,$ its velocity is reduced from $v_{0}$ to $\beta v_{0},$ where $\beta<1,$ to place the probe on a crash trajectory. Express in terms of $n$ and $\beta$ the angle $A O B$, where $B$ denotes the point of impact of the probe on the planet.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:10

Problem 115

A long-range ballistic trajectory between points $A$ and $B$ on the earth's surface consists of a portion of an ellipse with the apogee at point $C$. Knowing that point $C$ is $1500 \mathrm{~km}$ above the surface of the earth and the range $R \phi$ of the trajectory is $6000 \mathrm{~km}$, determine ( $a$ ) the velocity of the projectile at $C,(b)$ the eccentricity $e$ of the trajectory.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
10:13

Problem 116

A space shuttle is describing a circular orbit at an altitude of $563 \mathrm{~km}$ above the surface of the earth. As it passes through point $A,$ it fires its engine for a short interval of time to reduce its speed by $152 \mathrm{~m} / \mathrm{s}$ and begin its descent toward the earth. Determine the angle $A O B$ so that the altitude of the shuttle at point $B$ is $121 \mathrm{~km}$. (Hint: Point $A$ is the apogee of the elliptic descent trajectory.)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:12

Problem 117

As a spacecraft approaches the planet Jupiter, it releases a probe which is to enter the planet's atmosphere at point $B$ at an altitude of $280 \mathrm{mi}$ above the surface of the planet. The trajectory of the probe is a hyperbola of eccentricity $e=1.031 .$ Knowing that the radius and the mass of Jupiter are $44,423 \mathrm{mi}$ and $1.30 \times 10^{26}$ slug, respectively, and that the velocity $v_{B}$ of the probe at $B$ forms an angle of $82.9^{\circ}$ with the direction of $O A,$ determine $(a)$ the angle $A O B,(b)$ the speed $v_{B}$ of the probe at $B$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:10

Problem 118

A satellite describes an elliptic orbit about a planet. Denoting by $r_{0}$ and $r_{1}$ the distances corresponding, respectively, to the perigee and apogee of the orbit, show that the curvature of the orbit at each of these two points can be expressed as
$$\frac{1}{\rho}=\frac{1}{2}\left(\frac{1}{r_{0}}+\frac{1}{r_{1}}\right)$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:38

Problem 119

(a) Express the eccentricity $e$ of the elliptic orbit described by a satellite about a planet in terms of the distances $r_{0}$ and $r_{1}$ corresponding, respectively, to the perigee and apogee of the orbit. (b) Use the result obtained in part $a$ and the data given in Prob. $12.109,$ where $R_{E}=149.6 \times 10^{6} \mathrm{~km},$ to determine the approximate maximum distance from the sun reached by comet Hyakutake.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:50

Problem 120

Derive Kepler's third law of planetary motion from Eqs. (12.37) and (12.43)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:43

Problem 121

Show that the angular momentum per unit mass $h$ of a satellite describing an elliptic orbit of semimajor axis $a$ and eccentricity $e$ about a planet of mass $M$ can be expressed as
$$h=\sqrt{G M a\left(1-\varepsilon^{2}\right)}$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator