dynamics homework solutions

DYNAMICS Kinematics of Particles Chapter 11

1

Given:

Find:

t when v = 0

v, a and d when x = 0

Assumption:

(a)

Velocity is zero at time 6 s.

(b)


2

At

Distance traveled from



| | | |

When x = 0, ⁄ ⁄


3

11.11

At t = 0 s, v=16 in/s

∫

∫

At

⁄


4

At t = 1 s, x = 20 in

∫

∫

(

) (

)

At t = 7 s


⁄

(

)



| | | |


5

11.19

The packing material can be treated as a spring, and the equipment as a particle in simple harmonic

motion. Its velocity will be maximum when the particle passes through the equilibrium position (x = 0),

and its acceleration will be maximum at the turning points (first turning point at x = 20 mm).

While the box is falling, the packing material is not being compressed, so x = 0. Compression starts when

the box hits the ground at 4 m/s, so we will considered this velocity to be v0 , and x0 = 0 m. Final velocity

is vf = 0 m/s, and xf = 0.02 m. Maximum acceleration of the equipment is reached at the time of

maximum compression. Acceleration is given as a function of x. Positive x direction is up.

Given:

⁄ ⁄

Find:

Maximum acceleration of the equipment.

1. Find value of k.

∫

∫

∫

∫

(

) (

)

2. Use known values of k and xmax to find amax.

⁄ (upwards)


6

11.24

Given:

⁄

Find:

∫ ∫

| |

|

(

| |

)

(

| |

)

| |

√


7

11.27

(a)

∫

∫

[ ]

[ ]

⁄

⁄

(b)

[ ]

[ ]

(c)

⁄

⁄


8

11.28

Given:

Find:

a at x = 2 m when v0 = 3.6 m/s

t from x = 1 to x = 3 m when v0 = 3.6 m/s

(a)

(

)

(

) (

) (

)

For v0 = 3.6 m/s and x = 2:

(b)

(

)

∫

∫ [ ]

[ ]


9

11.40

Given:

⁄

⁄

Find: and time when runner B

should begin to run.

(a)

Runner A:

⁄

Velocity of runner A at t = 1.82 s = velocity of runner B

Runner B:

⁄

(b) Find how long it takes runner B to go from 0 m/s to 9.08 m/s:

Therefore, runner B should start running 2.59 seconds before runner A reaches the exchange zone.


10

11.45

Given:

⁄

From

From

⁄

At


11

At

( )

(

)

(

) (

)

(

)

( )

answer to (b)

(

)

⁄

(c)

(

)


12

11.61

Velocity:

⁄

⁄

⁄

⁄

⁄

⁄

⁄

Position:


13

(a)

v(t) vs t

x(t) vs t

(b)

⁄

11.101


14

Given: ⁄

Find:

Find time when

Find

(a)

Find time when


15

Find x when t = 1.2712 s:

(b) Therefore, the ball will land 7.01 m away from the net.


16

11.112

Given: ⁄

Find: Largest value (less than 45o) of the angle for which .

Time required for the puck to reach

Time when

Angle when ⁄ :

(

)

(

)

. At this angle, the puck will hit the crossbar.

(a) The largest value (less than 45o) of the angle for which the puck will enter the net is 14.66

o.

(b)


17

11.114

Given:

Find: Minimum value for .

(

)

(

)

√

According to the plot to the right of ,

has a minimum value between 0 and 900

at 2.53 m/s when

Therefore:


18

11.119

Given:

Velocity of boat:

Velocity of boat relative to river:


19

11.125

Given:

Find:

When

DYNAMICS Kinetics of Particles: Newton’s Second law Chapter 12

1

12.4

Given:

Load on spring scale:

Find: Weight of packages.

Load indicated by the spring scale and

the mass needed to balance the lever scale

when the elevator moves upward with an

acceleration of 4 ft/s2.

(a)

Resolve forces on y-axis:

∑

(

)

( )

𝑚 𝑊

𝑔

6

0 5 𝑠𝑙𝑢𝑔

The load on the spring (force 𝐹𝑠) is equal

to the mass of the package times the

acceleration of the elevator and the

acceleration due to gravity combined:

𝐹𝑠 𝑚𝑎 0 5 𝟏𝟒 𝟏 𝒍𝒃

when elevator is going down


2

(b)

∑

(

)

6 (

)

(Load indicated by the spring scale when the elevator accelerates

upwards.)

Therefore,

Mass of the weights = Mass of the package

6

𝑚 𝑊

𝑔

6

0 5 𝑠𝑙𝑢𝑔

The load on the spring (force 𝐹𝑠) is

equal to the mass of the package times

the acceleration of the elevator and

the acceleration due to gravity

combined:

𝐹𝑠 𝑚𝑎 0 5 𝟏𝟖 𝟏 𝒍𝒃

when elevator is going up

In order for the lever scale to be in equilibrium, the mass of the weights has to equal the

mass of the package. If the lever scale is in equilibrium, any effects caused by acceleration

will be equal on both sides of the scale, so they can be neglected.


3

12.7

Given:

60

50

Find:

x at

F.B.D. of bus on level road.

Forces acting on the bus:

Weight W

Normal N

Traction force P

Level:

∑


4

Uphill:

∑

(

)

F.B.D. of bus on the incline

(

)

0

0

0

66


5

12.10

Given:

⁄

Find:

F.B.D. of package at point A, with x-axis

along the slope:

Find an expression for the force of friction:

0

0 0

0

0

0

0 0


6

0 0

0

0 0

F.B.D. of package at point B:

0

5 0

5

5 0 5

5

5 0 5

5 0 5

5 0 5


7

12.13

Given: 0 0 0 0 5 66 0

Find: Shortest distance in which the rig can be brought to a stop if the load is not to shift.

F.B.D. of load while braking (load wants

to move fwd). Sliding is impending.

0

0

0

0

0

0 66

66


8

12.14

Given:

60

0

5 000

00

600 and 00

Find: (a) the distance traveled by the tractor-trailer before it comes to a stop.

(b) the horizontal component of the force in the hitch between the tractor and the

trailer while they are slowing down.

F.B.D. of tractor and trailer combined:

(a)

600 00 5 000 00

00

00

0


9

(b) Replace cab with a coupling. Assume a tensile force on the coupling. Force

is acting on the coupling.

00

00

00

00


10

12.22 To transport a series of bundles of shingles A to a roof, a contractor uses a motor-

driven lift consisting of a horizontal platform BC which rides on rails attached to the sides of

a ladder. The lift starts from rest and initially moves with a constant acceleration a1 as shown.

The lift then decelerates at a constant rate a2 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 a1 and the largest

allowable deceleration a2 if the bundle is not to slide on the platform.

Given: 0 0

Find: The largest allowable acceleration a1 and the largest allowable deceleration a2 if the

bundle is not to slide on the platform.

(a)When the bundle accelerates, the force of

friction on the platform is directed to the right:

F.B.D.

65

65

65

65 [1]

65

65

65

65


11

65

65

65 65

65 65

65 65

65 65

0 0

65 0 0 65

(b) When the bundle decelerates, the force of friction on the platform is directed to the

left:

Repeat part (a) changing the sign for a:

65

65 [2]

65

65 65

0 0

65 0 0 65


12

12.24 The propellers of a ship of weight W can produce a propulsive force F0; they produce

a force of the same magnitude but of opposite direction when the engines are reversed.

Knowing that the ship was proceeding forward at its maximum speed v0 when the engines

were put into reverse, determine the distance the ship travels before coming to a stop.

Assume that the frictional resistance of the water varies directly with the square of the

velocity.

Given:

0

Find: The distance the ship travels before coming to a stop.

Since the frictional force is not dependent on the ship’s weight, forces acting on the vertical

plane are omitted, and all calculations occur in the horizontal plane, with the positive

direction being the direction in which the ship is moving.

full steam ahead full steam reverse

When the ship is moving full steam ahead at maximum speed (a constant speed), its

acceleration is zero:

0

0

When the ship’s engines are put on reverse, the ship starts to decelerate until it comes to a

stop:


13

(

)

(

)

∫

∫

∫

∫

(

) [

]

(

)

(

)

(

)

(

)

(

) (

)


14

12.27 Determine the maximum theoretical speed that a 2700-

lb automobile starting from rest can reach after traveling a

quarter of a mile 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 D has a magnitude 0 0 , where D and v are

expressed in pounds and ft/s, respectively.

Given: 00 0 0 0

0 6 0 0 0

Find: The maximum theoretical speed.

F.B.D.

=

0 0 0 6 00

( )

0 0

00

6 0 6

00

0

0

0

∫

∫

0

As long as the traction tires are

not skidding, the force of friction

between them and the ground is

equal to the car’s force forward.

The car’s acceleration is a function

of velocity, therefore not constant.

Since the problem involves 𝑥, and

not 𝑡, integrate using 𝑎 𝑣 𝑑𝑣 𝑑𝑥 .


15

∫

∫ 0 000

∫

0 000∫

0 6

∫

0 000∫

650 5

0 000 650 5

5 000 650 5

5000 650 5

650 5

√

650 5

Insert value for :

√ 650 5


16

12.35 A 500-lb crate B is suspended from a cable attached to a 40-lb

trolley A which rides on an inclined I-beam as shown. Knowing that at

the instant shown the trolley has an acceleration of 1.2 ft/s2 up and to

the right, determine (a) the acceleration of B relative to A, (b) the

tension in cable CD.

Given: 500 0

Find: The acceleration of B relative to A, and the tension in cable CD.

F.B.D. of B

=

(a) For crate B:

0 5

5

5 5

(b) Find tension in cable AB:

5

5

(

5) 500 (

5) 50


17

(b)

F.B.D. of A

For trolley A:

5 5

5 5

50 5 0 5 0


18

12.37

Given: mass of ball 50 0 5

constant speed

Find:

F.B.D.

0

0

(

)

0 5

6

0 5

0 06

Solving for :

0 5

5


19

12.44 A child having a mass of 22 kg sits on a swing and is

held in the position shown by a second child. Neglecting the

mass of the swing, determine the tension in rope AB (a) while

the second child holds the swing with his arms outstretched

horizontally, (b) immediately after the swing is released.

Given: 5

Find: T when swing is not moving

T immediately after the swing is released F.B.D. when swing is held:

0

5 0

5

5 6

Swing is held by two ropes, so tension of each rope is:

6

F.B.D. when swing is released:

Immediately after the swing is released, the swing’s

acceleration is still zero:

0

5

5

5

6

Swing is held by two ropes, so tension of each rope is:

6


20

12.46 During a high-speed chase, a 2400-lb sports car

traveling at a speed of 100 mi/h just loses contact with

the road as it reaches the crest A of a hill. (a) Determine

the radius of curvature of the vertical profile of the

road at A. (b) Using the value of found in part a,

determine the force exerted on a 160-lb driver by the

seat of his 3100-lb car as the car, traveling at a constant

speed of 50 mi/h, passes through A.

Given: 00 00 6 6 60

50

Find:

The force exerted on a 160-lb driver by the seat of his 3100-lb car as the car, traveling

at a constant speed of 50 mi/h, passes through A.

When the car loses contact with the road at point A, Normal and frictional forces are zero:

F.B.D of Maserati Quattroporte as it loses contact with the road:


21

(b) Driver is not moving horizontally with respect to her seat, so 0.

F.B.D. of driver at point A

(

)

60 (

66 )


22

12.51 A curve in a speed track has a radius of 1000-ft and a rated

speed of 120 mi/h. (See Sample Prob. 12.6 for the definition of rated

speed.) Knowing that a racing car starts skidding on the curve when

traveling at a speed of 180 mi/h, determine (a) the banking angle θ, (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.

Given: radius of track 000

rated speed 0 6

skidding speed 0 6

Find: (a) the banking angle θ

(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.

=

(

)

The car travels in a horizontal circular

path of radius 𝜌. The normal component

𝑎𝑛 of the acceleration is directed toward

the center of the path; its magnitude is

𝑎𝑛 𝑣 𝜌, where 𝑣 is the speed of the car

in ft/s. Between 120 mph and 180 mph, 𝐹𝑓

is what keeps the car from skidding.


23

(a) At rated speed, 0

0 (

)

(

)

6

000 0 6

(

)

Substituting [1] and [2]:

(

)

(

)

6 000

6 000


24

(c) As speed decreases below 120 mph, Ff is what keeps the car from sliding down, so Ff

in this case is pointing in the opposite direction as the force of friction that kept the car from

sliding up the bank. Therefore,

Substituting [1] and [2]:

(

)

(

)

0 000

000

0 5

05 6

0 0 050 0 0 5

0

√

0 5 6


25

12.59 Three seconds after a polisher is started from rest, small

tufts of fleece from along the circumference of the 225-mm-

diameter polishing pad are observed to fly free of the pad. If the

polisher is started so that the fleece along the circumference

undergoes a constant tangential acceleration of 4 m/s2, 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

mg.

Given: 5 0 5 0 5

6 6 0

Find: (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 avg mass of a tuft is 1.6

mg.

F.B.D.

(a)

0

(b)

6 0 6 0

6 0 05

√

DYNAMICS Kinetics of Particles: Energy and Momentum Methods Chapter 13

1

13.5 Determine the maximum theoretical speed

that may be achieved over a distance of 360 ft by a

car starting from rest assuming there is no slipping.

The coefficient of static friction between the tires and

pavement is 0.75, and 60 percent of the weight of the

car is distributed over its front wheels and 40 percent

over its rear wheels. Assume (a) front-wheel drive,

(b) rear-wheel drive.

Given:

Find: Maximum theoretical speed.

Principle of work and energy:

Kinetic energy:

Position 1:

Position 2:

Work from 1 to 2:

( ) 𝑁𝑓 𝑊

𝐾 𝑈 𝐾

( 𝑊)( 𝑓𝑡) 𝑊

2𝑔𝑣

𝑣 ( )( )2( 2 2)

1 2 1 𝑓𝑡 𝑠 𝟔𝟗 𝟔 𝒎𝒑𝒉

(a) Front-wheel drive:

𝐹 𝐹𝑓 𝜇𝑁𝑓 𝑁𝑓

( ) 𝑁𝑟 𝑊

𝐾 𝑈 𝐾

( 𝑊)( 𝑓𝑡) 𝑊

2𝑔𝑣

𝑣 ( )( )2( 2 2)

8 𝑓𝑡 𝑠 𝟓𝟔 𝟗 𝒎𝒑𝒉

(b) Rear-wheel drive:

𝐹 𝐹𝑓 𝜇𝑁𝑟 𝑁𝑟


2

13.6 Skid marks on a drag race track indicate that the rear (drive) wheels of a car slip for

the first 60 ft of the 1320-ft track. (a) Knowing that the coefficient of kinetic friction is 0.60,

determine the speed of the car at the end of the first 60-ft portion of the track if it starts from

rest and the front wheels are just off the ground. (b) What is the maximum theoretical speed

for the car at the finish line if, after skidding for 60 ft, it is driven without the wheels slipping

for the remainder of the race? Assume that while the car is rolling without slipping, 60

percent of the weight of the car is on the rear wheels and the coefficient of static friction is

0.85. Ignore air resistance and rolling resistance.

Given: Point 1 at

Point 2 at

Point 3 at 1 2

8 rear-wheel drive

Find: and under given conditions.

𝐾

𝑈 𝐹𝑥 𝑊𝑥

( )𝑊( ) 𝑊

𝐾 1

2

𝑊

𝑔𝑣

𝑲𝟏 𝑼𝟏 𝟐 𝑲𝟐

𝑊 𝑊

2𝑔𝑣

𝑣 2𝑔 8 1 𝑓𝑡 𝑠 𝟑𝟐 𝟖 𝒎𝒑𝒉

(a) All weight on rear traction wheels.

Spinning means 𝐹 𝐹𝑓 𝜇𝑘𝑁𝑟 𝜇𝑘𝑊

𝐾 1

2

𝑊

𝑔 8 1 𝑊

𝑈 𝐹𝑥 𝜇𝑠 𝑊𝑥

( 8 )( )𝑊(12 )

2 𝑊

𝐾 1

2

𝑊

𝑔𝑣

𝑲𝟏 𝑼𝟏 𝟐 𝑲𝟐

𝑊 2 𝑊 𝑊

2𝑔𝑣

𝑣 1 𝑔 2 9 𝑓𝑡 𝑠 𝟏𝟒𝟐 𝟓 𝒎𝒑𝒉

(b) 𝐹 𝐹𝑓 𝜇𝑠𝑁𝑟 𝜇𝑠 𝑊

𝑥 1 2 − 12 𝑓𝑡


3

13.15 The subway train shown is traveling at a speed of 30 mi/h when the brakes are fully

applied on the wheels of cars B and C, causing them to slide on the track, but are not applied

on the wheels of car A. Knowing that the coefficient of kinetic friction is 0.35 between the

wheels and the track, determine (a) the distance required to bring the train to a stop, (b) the

force in each coupling.

Given:

8

1

8

Find: x and the force in each coupling.

(a)

(1

2)

(1

2)2

81 1 9

− ( ) −

81 1 9 −

81 1 9


4

(b) Force in coupling AB:

(1

2)

18

2

11 18

− ( ) 12 − (18 )12

12 − 81 8

11 18 12 − 81 8

2

12 19 82 ( )

Force in coupling BC:

(1

2)

8

2 2 9 9

−

12 − (8 )12

12 − 8

2 9 9 12 − 8

1 8 11

12 8 1 ( )

Replace wagon A with a coupling. Assume a tensile

force on the coupling. Force 𝐹𝐴𝐵 is acting on the coupling.

Replace wagons A and B with a coupling. Assume a tensile

force on the coupling. Force 𝐹𝐵𝐶 is acting on the coupling.


5

13.17 A trailer truck enters a 2 percent downhill grade traveling at 108 km/h and must slow

down to 72 km/h in 300 m. The cab has a mass of 1800 kg and the trailer 5400 kg. Determine

(a) the average braking force that must be applied, (b) the average force exerted on the

coupling between cab and trailer if 70 percent of the braking force is supplied by the trailer

and 30 percent by the cab.

Given:

1 1 8 18

1 8 2 2

Find: and force on coupling

(a)

(18 )9 81 2

1 12

1 12

1

2

1

2 2 ( ) 2

1

2

1

2 2 (2 ) 1

− 2 −

2 1 2 − 1 1

2 22 1


6

(b)

( )9 81 29

1 9 8

1 9 8

1

2

1

2 ( ) 2

1

2

1

2 (2 ) 1 8

− 1 8 − 1 2 − 1 2 8

2 1 − 1 2 8 1 8 1

−111 2

−111 2

− 1

Replace cab with a coupling. Assume a tensile force

on the coupling. Force 𝐹𝐴𝐵 is acting on the coupling.


7

13.27 A 10-lb block is attached to an unstretched

spring of constant 12 . The coefficients of

static and kinetic friction between the block and the

plane are 0.60 and 0.40, respectively. If a force F is

applied to the block until the tension in the spring reaches 20 lb and then suddenly removed,

determine (a) how far the block will move to the left before coming to a stop, (b) whether the

block will then move back to the right.

Given: 1 − −2 12

Find: (a) how far the block will move to the left before coming to a stop,

(b) whether the block will then move back to the right.

1. Find how far to the right the spring is stretched by force F:

− −2 −2

−12 1

2. Draw F.B.D. with forces and points:

Point 1: Block is released from rest after spring has been stretched by force F.

Point 2: Block stops after spring has been compressed to its maximum.

Point 3: Block possibly moves back to the right.

The force on the spring is 20 lb, therefore the reaction force by the spring is -20 lb.

The work done by a spring is

negative when the spring is being

stretched or compressed, and

positive when it is released.


8

From point 1 to point 2:

−

1

2 (

− ) − ( − )

1

212(1 −

) − ( )(1 )(1 − )

− − 1

(a)

− − 1

− 1

Therefore, the block moves from +1.667 to –1, a distance of 2.667 in or 0.222 ft.

(b) When the block stops at point 2, it will move again to the right if the force exerted by

the spring is greater than the static frictional force exerted by the surface on which the block

stands.

− −(12)(−1) 12

( )(1 )

, therefore the block will move back to the right.


9

13.28 A 3-kg block rests on top of a 2-kg block supported by but not

attached to a spring of constant 40 N/m. The upper block is suddenly

removed. Determine (a) the maximum speed reached by the 2-kg block, (b)

the maximum height reached by the 2-kg block.

Given: 2

Find: (a) the maximum speed reached by the 2-kg block,

(b) the maximum height reached by the 2-kg block.

1. Find the force on the spring by the weight of the masses:

( 2)(−9 81) − 9

2. Find the distance that a weight of 49.05 N will compress the given spring:

− −

9

−1 22

3. Define points:

Point 1: 3-kg block has been removed. Speed of 2-kg block is zero and its position is

−1 22 .

Point 2: Spring and block reach their maximum speed at .

Point 3: Spring has been left behind and block reaches its maximum height.

When the 3-kg block is removed, the spring will stretch a certain

distance while pushing the 2-kg block. At x = 0, the spring and the block

will reach their maximum speed. At that point the spring will slow

down until its speed is zero at its maximum stretch, and the 2-kg block

will keep going upwards until the force of gravity makes it reach a

speed of zero before it makes it fall down.


10

4. Apply the method of work and energy to find maximum velocity of the spring-mass

system at

1

2

1

22

−

1

2 (

− ) − ( )

1

2 (−1 22 − ) − (2)(9 81)(1 22 )

2 − 2

1 2

1 2

5. Apply the method of work and energy to find height of block at point 3.

−

1

2 (

− ) − ( )

1

2 (−1 22 − ) − (2)(9 81)( ) 2 − 19 2

2 − 19 2

2

19 2 1 29

The spring stops doing

work on the block at y = 0.


11

13.44 A small block slides at a speed 8 on a

horizontal surface at a height above the ground.

Determine (a) the angle θ at which it will leave the cylindrical

surface BCD, (b) the distance x at which it will hit the ground.

Neglect friction and air resistance.

Given: 8

Find: and

At point B:

[speed is constant]

−

At point C, using n and t coordinates:

( )

1

2

1

2 8 2

1

2

1

2

( − ) ( − )

2 ( − ) 1

2

2 ( − ) 1

2


12

2 2 2( − ) 1

2 2 2

2 9 − 2 2 1 1

8 128

2 2

2

88

88

Find velocity at point C:

2 ( − ) 1

2

2 ( − ) 1

2 2 2 2( − 2 )

1

2

2 2 2( − 2 ) 1

2

9 2

8 21 2

−

2 2 2 −1

2 2 2

1 1 − 2 − 2 2

(8 21 )( )

2


13

13.46 (a) A 120-lb woman rides a 15-lb bicycle up a 3-percent slope at a constant speed of

5 ft/s. How much power must be developed by the woman? (b) A 180-lb man on an 18-lb

bicycle starts down the same slope and maintains a constant speed of 20 ft/s by braking. How

much power is dissipated by the brakes? Ignore air resistance and rolling resistance.

Given: 12 1 1 18 18 198

1 18

(a) Forces in the direction of v: −1

(−1 )( )

(a) Forces in the direction of v: 198

(198 )(2 )


14

13.49 In an automobile drag race, the rear (drive) wheels of a l000-kg car skid for the first

20 m and roll with sliding impending during the remaining 380 m. The front wheels of the

car are just off the ground for the first 20 m, and for the remainder of the race 80 percent of

the weight is on the rear wheels. Knowing that the coefficients of friction are 9 and

8, determine the power developed by the car at the drive wheels (a) at the end of the

20-m portion of the race, (b) at the end of the race. Give your answer in kW and in hp. Ignore

the effect of air resistance and rolling friction.

Given: 1 9 8

Find: (a) The power developed by the car at the drive wheels at the end of the 20-m portion

of the race,

(b) at the end of the race

(a) All weight on rear traction wheels.

Spinning means ( 8)(1 )(9 81) 8

( 8)(2 ) 1 1

1

2

1

21

1 1

√1 1

1

( 8 )(1 ) 1 8 9

1 kW = 1.34102209 hp


15

(b) Roll with sliding impending means:

8 ( 9 )( 8)(1 )(9 81) 2

8

1

2 ( )(1 ) 1 1

( 2)( 8 ) 2 8 1

1

2

1 1 2 8 1

√ 9

( 2)( ) 2 11


16

13.69 A spring is used to stop a 50-kg package

which is moving down a 20° incline. The spring

has a constant and is held by

cables so that it is initially compressed 50 mm.

Knowing that the velocity of the package is 2

m/s when it is 8 m from the spring and the

kinetic coefficient of friction between the

package and the incline is 0.2., determine the

maximum additional deformation of the spring

in bringing the package to rest.

Given: 2

Find: The maximum additional deformation of the spring in bringing the package to rest.

1

2

( )( )(2) 1

1

2

( )( ) 2

− ( 2 − 2 )

( 2 − 2 )

(8)( )(9 81)( 1 1) 2

1 2 2 9

2 2 ( 9) 2

2 ( )

1 ( )

1

2 ( )

1 ( )

2 1 ( ) 1 ( )

1 ( ) − 1 ( ) − 2 12

( ) 2281 ( ) 2281


17

13.144 The design for a new cementless hip implant is to be studied

using an instrumented implant and a fixed simulated femur. Assuming

the punch applies an average force of 2 kN over a time of 2 ms to the

200 g implant, determine (a) the velocity of the implant immediately

after impact, (b) the average resistance of the implant to penetration if

the implant moves 1 mm before coming to rest.

Given: 2 ( − ) 2 2

Find: (a) the velocity of the implant immediately after impact,

(b) the average resistance of the implant to penetration

if the implant moves 1 mm before coming to rest.

( − ) (2 )( 2)

2

2

1

2

( )( 2)(2 ) 2

− − 1

2 − 1


18

13.174 A 1-kg block B is moving with a velocity v0 of

magnitude 2 as it hits the 0.5-kg sphere A,

which is at rest and hanging from a cord attached at O.

Knowing that between the block and the

horizontal surface and 8 between the block and the

sphere, determine after impact (a) the maximum height h

reached by the sphere, (b) the distance x traveled by the

block.

Given: 1

2 8

Find: (a) the maximum height h reached by the sphere,

(b) the distance x traveled by the block.

(1)(2) 1 2

− −

−

−2 8

− −1

1 2} [

−1 1 −1 1 2

] [1 2 1 8

] 2 8

(a) Sphere:

1

2

( )( )(2 ) 1

( )(9 81) 9

1 9


19

(b) Block:

1

2

( )(1)( 8) 2

− − ( )(1)(9 81) − 88

2 − 88


20

13.180 Two cars of the same mass run head-on into each other at C. After the collision, the

cars skid with their brakes locked and come to a stop in the positions shown in the lower part

of the figure. Knowing that the speed of car A just before impact was 5 mi/h and that the

coefficient of kinetic friction between the pavement and the tires of both cars is 0.30,

determine (a) the speed of car B just before impact, (b) the effective coefficient of restitution

between the two cars.

Given:

−12

−

Find:

(a) the speed of car B just before

impact,

(b) the effective coefficient of restitution between the two cars.

1. Conservation of linear momentum:

[general form]

𝐾 1

2𝑚𝑣 𝐾

𝑈 −𝐹𝑓𝑥 −𝜇𝑘𝑁𝑥

𝑈 −( )𝑚( 2 2)(12) 11 92 𝑚

𝑲𝟏 𝑼𝟏 𝟐 𝑲𝟐

1

2𝑚𝑣 − 11 92 𝑚

𝑣𝐴 (2)(11 92) 1 22 𝑓𝑡 𝑠

CAR A

𝐾 1

2𝑚𝑣 𝐾

𝑈 −𝐹𝑓𝑥 −𝜇𝑘𝑁𝑥

𝑈 −( )𝑚( 2 2)( ) 28 98 𝑚

𝑲𝟏 𝑼𝟏 𝟐 𝑲𝟐

1

2𝑚𝑣 − 28 98 𝑚

𝑣𝐵 (2)(28 98) 1 1 𝑓𝑡 𝑠

CAR B


21

− − − [masses are equal, ]

(a)

1 22 1 1 2

(b)

− −

− 1 1 − (−1 22 )

− (− 1 2 )

1 2

1


22

Class Example

Vehicles A and B collide as shown.

Given: 1

Find: final position (brakes applied,

both vehicles skid).

1. TANGENTIAL AXIS

Tangential velocities don’t change after impact:

( ) ( ) 2 2 2

( ) ( )

2. NORMAL AXIS

( ) ( ) ( ) ( ) [B has no normal velocity]

( 2 )

( )

( )

( 2 )

( )

1

( )

( 2 ) ( ) 1 ( ) [g cancelled]

( 2 ) ( ) ( ) [divide both sides by 1000]

( ) 2 2 − ( ) [1 equation, 2 unknowns]

3. USE e TO FIND ANOTHER EQUATION WITH SAME VARIABLES

− −

(

) − ( )

( ) − ( )

( ) − ( )

− 2

( ) − ( ) − 2 (

) ( ) 2

2 2 − ( ) ( ) 2 (

) 21 2


23

( ) 2 2 9 ( ) 2 2 2

( ) 2 2 − ( 2 2 ) 2 2 ( )

4. FIND RESULTING VELOCITIES:

9 2 2 −

−

5. WITH INITIAL VELOCITIES AFTER COLLISION, FINAL VELOCITIES

(ZERO), AND WORK DONE (FRICTION), THE WORK AND ENERGY METHOD CAN

BE USED TO FIND FINAL POSITIONS.

DYNAMICS Systems of Particles Chapter 14

1

14.3 A 180-lb man and a 120-lb woman stand side by side

at the same end of a 300-lb boat, ready to dive, each with a

16-ft/s velocity relative to the boat. Determine the velocity of

the boat after they have both dived, if (a) the woman dives

first, (b) the man dives first.

Given:

Find: The velocity of the boat after they have both dived, if (a) the woman dives first, (b)

the man dives first.

Principle of Conservation of Linear Momentum L:

(a)

L before anyone dives = L after woman dives:

(

)

( )

L after woman dives = L after man dives

(

) ( )

( )

𝑣𝑑 𝑣𝑏 𝑣𝑑 𝑏

𝑣𝑑 𝑣𝑏

𝑣𝑑 𝑣𝑏


2

(b)

L before anyone dives = L after man dives:

(

)

( )

L after man dives = L after woman dives

(

) ( )

( )


3

14.5 A bullet is fired with a horizontal velocity of

1500 ft/s through a 6-lb block A and becomes

embedded in a 4.95-lb block B. Knowing that blocks A

and B start moving with velocities of 5 ft/s and 9 ft/s,

respectively, determine (a) the weight of the bullet, (b)

its velocity as it travels from block A to block B.

Given:

Find: and between A and B

(a)

L of bullet = L of block A + L of block B and bullet

( )

1491

(b)

Initial L of bullet = L of bullet after going through block A

( ) ( )

( )

( )

( )


4

14.8 At an amusement park there are 200-kg bumper cars A, B, and C that have riders with

masses of 40 kg, 60 kg, and 35 kg respectively. Car A is moving to the right with a velocity

vA = 2 m/s when it hits stationary car B. The coefficient of restitution between each car is 0.8.

Determine the velocity of car C so that after car B collides with car C the velocity of car B is

zero.

Given:

Find: so that after car B collides with car C, the velocity of car B is zero.

( )( ) ( )( )

[

] [

]

( )( ) ( )( )

[

] [

]


5

14.17 A small airplane of mass 1500 kg and a helicopter of mass 3000 kg flying at an

altitude of 1200 m are observed to collide directly above a tower located at O in a wooded

area. Four minutes earlier the helicopter had been sighted 8.4 km due west of the tower and

the airplane 16 km west and 12 km north of the tower. As a result of the collision the

helicopter was split into two pieces, H1 and H2, of mass m1 = 1000 kg and m2 = 2000 kg,

respectively; the airplane remained in one piece as it fell to the ground. Knowing that the two

fragments of the helicopter were located at points H1 (500 m, –100 m) and

H2 (600 m, –500 m), respectively, and assuming that all pieces hit the ground at the same

time, determine the coordinates of the point A where the wreckage of the airplane will be

found.

Given:

( ) ( )

( ) ( )

Find: ( )

1. Find velocity of airplane and helicopter at time of collision:

( ) ( )

( )

( ) ( )

( )

2. Find velocity of mass center G of the fragments after the collision:

( )

( )( ) ( )( ) ( )

The mass center G

of a system of

particles moves as

if the entire mass of

the system and all

the external forces

were concentrated

at that point.


6

3. Find the time it takes the fragments to fall freely from to

√

√

4. Find position of G at time of impact with the ground:

( )

5. Use Equation 14.12 to find position of plane where it hits the ground:

∑

( ) ( ) ( )

( )( ) ( ) ( )( ) ( )( )

( )( ) ( ) ( )( ) ( )( )

( ) ( ) ( )

( )

( )

( ) ( )


7

14.19 Car A was traveling east at high speed when it collided at point O with car B, which

was traveling north at 72 km/h. Car C, which was traveling west at 90 km/h, was 10 m east

and 3 m north of point O at the time of the collision. Because the pavement was wet, the

driver of car C could not prevent his car from sliding into the other two cars, and the three

cars, stuck together, kept sliding until they hit the utility pole P. Knowing that the masses of

cars A, B, and C are, respectively, 1500 kg, 1300 kg, and 1200 kg, and neglecting the forces

exerted on the cars by the wet pavement solve the following problem:

Knowing that the coordinates of the utility pole are xp = 18 m and yp = 13.9 m, determine (a)

the time elapsed from the first collision to the stop at P, (b) the speed of car A.

Given:

( )

Find:

Time from O to and ( )

Use Equation 14.12 to work backwards from final position:

∑

( )

( )( ) ( )

( )( ) ( ) ( )

( ) ( )


8

Equating coefficients of :

Equating coefficients of :

( )

( )


9

14.73 A floor fan designed to deliver air at a maximum

velocity of 6 m/s in a 400-mm-diameter slipstream is supported

by a 200-mm-diameter circular base plate. Knowing that the

total weight of the assembly is 60 N and that its center of

gravity is located directly above the center of the base plate,

determine the maximum height h at which the fan may be

operated if it is not to tip over. Assume ρ = 1.21 kg/m3 for air

and neglect the approach velocity of the air.

Given:

Find: Maximum height h at which the fan may be operated if it is not to tip over.

1. Find mass flow rate dm/dt

( ) (

( )

)

2. Find force of the fluid Ff

( ) ( )( )

3. Find force of thrust Fth

4. Sum forces on an axis or moments.

In this case, we will sum moments about the center of gravity in line with the base of the fan.

When the fan is in “impending tip”, the normal force N is applied at the “tipping hinge”.

( )

( )( )

𝑑𝑚

𝑑𝑡 𝜌𝑄

𝜌𝐴𝑣

𝛾

𝑔𝑄

𝛾

𝑔𝐴𝑣

Mass flow rate


10

14.74 The helicopter shown can produce a maximum

downward air speed of 80 ft/s in a 30-ft-diameter slipstream.

Knowing that the weight of the helicopter and its crew is 3500

lb and assuming lb/ft3 for air, determine the

maximum load that the helicopter can lift while hovering in

midair.

Given:

(specific weight of air)

Find: The maximum load L that the helicopter can lift while hovering in midair.

1. Find mass flow rate dm/dt

(

)

2. Find force of the fluid Ff

( ) ( )( )

3. Find force of thrust Fth

4. Sum forces on an axis or moments.

[acceleration is zero when hovering]


11

14.75 A jet airliner is cruising at a speed of 600 mi/h with each of its three engines

discharging air with a velocity of 2000 ft/s relative to the plane. Determine the speed of the

airliner after it has lost the use of (a) one of its engines, (b) two of its engines. Assume that

the drag due to air friction is proportional to the square of the speed and that the remaining

engines keep operating at the same rate.

Given:

Find: The speed of the airliner after it has lost the use of

(a) one of its engines,

(b) two of its engines.

( )

[acceleration is zero when at cruising speed]

( )

With three engines running:

(

)

( )

(

)

With two engines running:

(

)

(

)

(

)

( )

( )

𝑣𝑖𝑛 𝑣𝑎𝑖𝑟𝑝𝑙𝑎𝑛𝑒

Moving reference frame:


12

With one engine running:

(

)

(

)

(

)

( )

( )

Dynamics Kinematics of Rigid Bodies Chapter 15

1

15.48 In the planetary gear system shown, the radius of gears

A, B, C, and D is 3 in. and the radius of the outer gear E is 9 in.

Knowing that gear E has an angular velocity of 120 rpm

clockwise and that the central gear has an angular velocity of

150 rpm clockwise, determine (a) the angular velocity of each

planetary gear, (b) the angular velocity of the spider connecting

the planetary gears.

Given:

Find:

Gear A: [1]

Spider: [2]

Gear B: [3]

[4]

Gear E: [5]

[ ]

[ ]

( )( ) ( )( )

[ ] ( )( )

( )

[ ]

( )


2

15.57 In the engine system shown, l = 160 mm and b = 60 mm.

Knowing that the crank AB rotates with a constant angular velocity of

1000 rpm clockwise, determine the velocity of the piston P and the

angular velocity of the connecting rod when (a) , (b) .

Given:

Find:

(a)

( )( )

( )

⁄

⁄ ⁄

( )

(b)

( )( )

( ) ( ) ( ) ( )

( )

⁄ ( )

( )


3

15.111 An automobile travels to the left at a

constant speed of 48 mi/h. Knowing that the diameter

of the wheel is 22 in., determine the acceleration (a) of

point B, (b) of point C, (c) of point D.

Given:

Find:

The wheel is rolling and not sliding, therefore point C is the instantaneous center.

( )( )

( )

( )

( )


4

15.125 Knowing that crank AB rotates about point A with a constant

angular velocity of 900 rpm clockwise, determine the acceleration of

the piston P when

Given:

Find:

Rod AB:

( )( )

( ) ( )( )

Rod BD:

( ) ( ) ( )

( )

( ) ( )

( ) ( )( )

( )( )

( ) ( )

Horizontal components

( )

Vertical components

( )( )

DYNAMICS Plane Motion of Rigid Bodies: Forces and Acceleration Chapter 16

1

16.7 A 20-kg cabinet is mounted on casters that allow it to move

freely ( ) on the floor. If a 100-N force is applied as shown,

determine (a) the acceleration of the cabinet, (b) the range of values of h

for which the cabinet will not tip.

Given:

Find: (a) the acceleration of the

cabinet, (b) the range of values of h for

which the cabinet will not tip.

When tipping is impending

When tipping is impending

Therefore, the range of values of h for which the cabinet will not tip is


2

16.33 In order to determine the mass moment of inertia of a flywheel of

radius 600 mm, a 12-kg block is attached to a wire that is wrapped around

the flywheel. The block is released and is observed to fall 3 m in 4.6 s. To

eliminate bearing friction from the computation, a second block of mass

24 kg is used and is observed to fall 3 m in 3.1 s. Assuming that the

moment of the couple due to friction remains constant, determine the

mass moment of inertia of the flywheel.

Given: When

When

Find: the mass moment of inertia of the flywheel.

Case 1:

Case 2:


3

16.62 The 3-oz yo-yo shown has a centroidal radius of gyration of 1.25 in.

The radius of the inner drum on which a string is wound is 0.25 in. Knowing

that at the instant shown the acceleration of the center of the yo-yo is 3 ft/s2

upward, determine (a) the required tension T in the string, (b) the

corresponding angular acceleration of the yo-yo.

Given: ⁄

⁄

⁄

⁄

Find: (a) the required tension T in the string,

(b) the corresponding angular acceleration of the yo-yo.

(

)


4

16.69 A bowler projects an 8-in.-diameter ball weighing 12 lb along an

alley with a forward velocity v0 of 15 ft/s and a backspin of 9 rad/s.

Knowing that the coefficient of kinetic friction between the ball and the

alley is 0.10, determine (a) the time t1 at which the ball will start rolling

without sliding, (b) the speed of the ball at time t1, (c) the distance the

ball will have traveled at time t1.

Given:

At the instant , the ball starts to roll,

point C becomes the instantaneous center of

rotation, and .

At

Kinematics at


5

(

)r

(

)


6

16.82 A turbine disk of mass 26 kg rotates at a constant rate of

9600 rpm. Knowing that the mass center of the disk coincides

with the center of rotation O, determine the reaction at O

immediately after a single blade at A, of mass 45 g, becomes loose

and is thrown off.

Given:

Find: the reaction at O immediately after a single blade at A, of mass 45 g, becomes loose

and is thrown off.


7

16.97 A homogeneous sphere S, a uniform cylinder C,

and a thin pipe P are in contact when they are released

from rest on the incline shown. Knowing that all three

objects roll without slipping, determine, after 4 s of

motion, the clear distance between (a) the pipe and the

cylinder, (b) the cylinder and the sphere. (SP 16.8)

Given:

Find: The clear distance, after 4 s of

motion, between (a) the pipe and the

cylinder, (b) the cylinder and the

sphere.

General case:

For pipe:

For cylinder:

For sphere:

(a)

(b)


8

16.104 A drum of 60-mm radius is attached to a disk of

120-mm radius. The disk and drum have a total mass of 6

kg and a combined radius of gyration of 90 mm. A cord is

attached as shown and pulled with a force P of magnitude

20 N. Knowing that the disk rolls without sliding,

determine (a) the angular acceleration of the disk and the

acceleration of G, (b) the minimum value of the

coefficient of static friction compatible with this motion. (See Sample Problem 16.9)

Given:

Find: (a) the angular acceleration of the disk and the acceleration of G,

(b) the minimum value of the coefficient of static friction compatible with this motion.

DYNAMICS Plane Motion of Rigid Bodies: Energy and Momentum Methods Chapter 17

1

17.9 Each of the gears A and B has a mass of

2.4 kg and a radius of gyration of 60 mm, while

gear C has a mass of 12 kg and a radius of

gyration of 150 mm. A couple M of constant

magnitude 10 N·m is applied to gear C.

Determine (a) the number of revolutions of gear

C required for its angular velocity to increase

from 100 to 450 rpm, (b) the corresponding

tangential force acting on gear A.

Given:

Find: (a) the number of revolutions of gear C required for its angular velocity to increase

from 100 to 450 rpm,

(b) the corresponding tangential force acting on gear A.

All three gears are in mesh, so their contact velocity is the same:

Moment of inertia of the gears:


2

Find kinetic energy of system at position 1:

(

)

(

)

(

)

[

]

Find kinetic energy of system at position 2:

(

)

(

)

(

)

[

]

Find work from position 1 to position 2 (work of the couple):

From the Principle of Conservation of Energy:

(a) Number of revolutions in 39.898 radians:

(b) Gear A:

(

)

(

)

[ ]

[ ]


3

17.24 A 20-kg uniform cylindrical roller, initially at

rest, is acted upon by a 90-N force as shown. Knowing

that the body rolls without slipping, determine (a) the

velocity of its center G after it has moved 1.5 m, (b) the

friction force required to prevent slipping.

Given:

Find: (a) the velocity of its center G after it has moved 1.5 m,

(b) the friction force required to prevent slipping.

[ ]

(

) (

)

√


4

17.69 A wheel of radius r and centroidal radius of gyration is

released from rest on the incline shown at time t = 0. Assuming that the

wheel rolls without sliding, determine (a) the velocity of its center at

time t, (b) the coefficient of static friction required to prevent slipping.

The external forces W, F, and N form a system equivalent to the system of effective forces

represented by the vector maG and the couple IGα.

No sliding means aG = r α

(a)


5

(b)

(

)

(

)

(

)

(

)

(

)


6

17.99 A 45-g bullet is fired with a velocity of 400 m/s at

θ = 5° into a 9-kg square panel of side b = 200 mm.

Knowing that the panel is initially at rest, determine (a)

the required distance h if the horizontal component of the

impulsive reaction at A is to be zero, (b) the

corresponding velocity of the center of the panel

immediately after the bullet becomes embedded.

Given:

Find: (a) the required distance h if the horizontal component of the impulsive reaction at A

is to be zero,

(b) the corresponding velocity of the center of the panel immediately after the bullet

becomes embedded.

F.B.D

Apply Principle of Impulse and Momentum for the plane motion of a rigid body:

Syst Momenta1 + Syst Ext Imp1→2 = Syst Momenta2

Moments about A :


7

[ ]

If the horizontal component of the impulsive reaction at A is to be zero:

Value of into equation [1]:

(a)

(b)

dynamics homework solutions

Documents

velocity of runner b

s distance

vt vs t xt vs t b

final velocity

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value of

positive x direction

time of maximum compression