02-ENGINEERING DYNAMICS (Worksheets) - (V 2558)

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6

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(Engineering Dynamics) " (Engineering Statics) " Engineering Mechanics DYNAMICS SI version Sixth edition J.L. Meriam L.G. Kraige John Wiley & Sons

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............................................................................................................................................... ...........................................................................................................................................

2 (Kinematics of Particles) 2. ......................................................................................... 2- 2. ........................................................................ 2-! 2.2.1 (x-y) ........................................................ 2-! 2.2.2 ( (n-t) ................................... 2-+ 2.2.3 (r-) .................................................... -/ 2.0 ....................................................................................... 2-+ 2./ ( ....................................................................... 2-! 3 (Kinetics of Particles) 3A (Force, Mass and Acceleration) .................... 0A-1 3A.1 ........................................................................... 0A-1 3A.2 ........................................................... 0A- 3B (Work and Energy) ................................................. 0B-1 3C (Impulse and Momentum) ............................. 0C-1 3C.1 ....................................................... 0C-1 3C.2 .......................................................... 0C-3 3C.3 ................................................................................................ 0C-1+ 4 (Kinetics of Systems of Particles) ....................................... /-

()

5 (Kinetics of Systems of Particles) ....................................... !- 5. ............................................................................................ !- 5. ....................................................................................... !-> 5.0 ..................................................................................................... !-1> 5.4 ............................................................................................ !- 5.! ..................................................................................................... !-2! 5.! ........................................................ !-0 6 (Kinetics of Rigid bodies) 6A (Force, Mass and Acceleration) .................... BA-1 6A.1 ....................................................... BA-2 6A.2 ....................................................... BA-! 6A.3 ......................................................... BA-F 6A.4 ............................................................ BA-0 6B (Work and Energy) ................................................. BB-1 6B.1 ....................................................................... BB-1 6B.2 ......................................................... BB- 6B.3 ............................................................ BB-/ 6C (Impulse and Momentum) ............................. BC-1 TABLE D/4 PROPERTIES OF HOMOGENEOUS SOLIDS ........................................... - .................................................................................................................................... -

2

[Kinematics of Particles]

Chapter 2 Kinematics of Particles Rectilinear Motion (1D)

2-1

. (Rectilinear Motion)

1: ([1] Problem 2/4)

The displacement of a particle which moves along the s-axis is given by s = (-2+3t)e-0.5t, where s is in

meters and t is in seconds. Plot the displacement, Velocity, and acceleration versus time for the first 20

seconds of motion. Determine the time at which at which the acceleration is zero.

Ans. t = 4.67 s


2-2

2: ([1] Problem 2/22)

A vehicle enters a test section of straight road at s = 0 with

a speed of 40 km/h. It then undergoes an acceleration which

varies with displacement as shown. Determine the velocity v of

the vehicle as it passes the position s = 0.2 km.

Ans. v = 20.1 m/s (72.3 km/h)


2-3

3: ([1] Problem 2/32)

A motorcycle patrolman starts from rest at A two seconds after a car, speeding at the

constant rate of 120 km/h, passes point A. If the patrolman accelerates at the rate of 6 m/s2 until

he reaches his maximum permissible speed of 150 km/h, which he maintains, calculate the

distance s from point A to the point at which he overtake the car.

Ans. s = 912 m


2-4

4: ([1] Problem 2/51)

When the effect of aerodynamic drag is included, the y-

acceleration of a baseball moving vertically upward is au = g

kv2, while the acceleration when the ball is moving downward is

ad = g + kv2, where k is a positive constant and v is the speed in

meters per second. If the ball is thrown upward at 30 m/s from

essentially ground level, compute its maximum height h and its

speed vf upon impact with the ground. Take k to be 0.006 m-1 and

assume that g is constant.

Ans. h = 36.5 m, vf = 24.1 m/s

Chapter 2 Kinematics of Particles Plane Curvilinear Motion (2D): x-y Coordinates

2-5

. (Plane Curvilinear Motion)

.. (x-y)

1: ([1] Problem 2/72)

With what minimum horizontal velocity u can a boy throw a

rock at A and have it just clear the obstruction at B?

Ans. umin = 28.0 m/s


2-6

2: ([1] Problem 2/74)

Water issues from the nozzle at A, which is 1.5 m above

the ground. Determine the coordinates of the point of impact

of the steam if the initial water speed is (a) v0 = 14 m/s and v0

= 18 m/s.

Ans. The coordinates (x,y): (a) (9 m, 8.98 m)

(b) (29.43 m, 0 m)


2-7

3: ([1] Problem 2/79)

A projectile is launched from point A with the

initial conditions shown in the figure. Determine the

slant distance s which locates the point B of impact.

Calculate the time of flight t.

Ans. s = 1,057 m, t = 19.50 s


2-8

4: ([1] Problem 2/94)

The baseball player likes to release his foul shots at an angle

= 50o to the horizontal as shown. What the initial speed v0 will

cause the ball to pass through the center of the rim?

Ans. v0 = 7.01m/s

Chapter 2 Kinematics of Particles Plane Curvilinear Motion (2D): r- Coordinates

2-9

.. G (n-t)

1: ([1] Problem 2/105)

The car travels at a constant speed from the bottom A of the dip to the top B of the hump. If the radius

of curvature of the road at A is A = 120 m and the car acceleration at A is 0.4g, determine the car speed v. If

the acceleration at B must be limited to 0.25g, determine the minimum radius of curvature B of the road at

B.

Ans. v = 21.6 m/s, B = 190.4 m


2-10

2: ([1] Problem 2/109)

The figure shows two possible paths for negotiating an unbanked

turn on a horizontal portion of a race course. Path A-A follows the

centerline of the road and has a radius of curvature A = 85 m, while

path B-B uses the width of the road to good advantage in increasing

radius of curvature to B = 200 m, If the drivers limit their speed in

their curves so that the lateral acceleration does not exceed 0.8g,

determine the maximum speed for the each path.

Ans. vA = 25.8 m/s, vB = 39.6 m/s


2-11

3: ([1] Problem 2/110)

Consider the polar axis of the earth to be fixed in the space and

compute the magnitude of acceleration av

of a point P on the earth surface

at latitude 40o north. The mean diameter of the earth is 12,742 km and its

angular velocity is 0.729(10-4) rad/s.

Ans. a = 0.0259 m/s2


2-12

4: ([1] Problem 2/113)

A space shuttle which moves in a circular orbit around the earth at a

height h = 240 km above its surface must have a speed of 27,955 km/h.

Calculate the gravitational acceleration g for this altitude. The mean

radius of the earth is 6,371 km. (Check your answer by computing g from

the gravitational law 2

0

Rg g

R h

= +

where g0 = 9.821 m/s2)

Ans. an = 0.0259 m/s2


2-13

5: ([1] Problem 2/116)

A car travels along the level curved road with a speed which is decreasing at the constant rate of 0.6 m/s

each second. The speed of the car as it passes point A is 16 m/s. Calculate the magnitude of the total

acceleration of the car as it passes point B which is 120 m along the road from A. The radius of the curvature

of the road at B is 60 m.

Ans. an = 1.961 m/s2


2-14

..K (r-)

1: ([1] Problem 2/135)

A car P travels along a straight road with a constant speed

v = 100 km/h. At the instant when the angle = 60o, determine

the values of r& in m/s and & in deg/s

Ans. r& = 13.89 m/s, & = -39.8 deg/s


2-15

2: ([1] Problem 2/138)

A model airplane flies over an observer O with constant speed in a straight line as shown. Determine

the signs (plus, minus, or zero) for , , , , ,r r r && && and && for each of the position A, B, and C.


2-16

3: ([1] Problem 2/147)

The rocket is fired vertically and tracked by the radar station

shown. When reaches 60o, other corresponding measurements

give the values r = 9 km, r&& = 21 m/s2, and & = 0.02 rad/s.

Calculate the magnitudes of the velocity and acceleration of the

rocket at this position.

Ans. v = 360 m/s, a = 20.1 m/s2


2-17

4: ([1] Problem 2/149)

A jet plane flying at a constant speed v at an altitude h = 10 km

is being tracked by radar located at O directly below the line of

flight. If the angle is decreasing at the rate of 0.020 rad/s when =

60o, determine the value of r&& at this instant and the magnitude of

the velocity vv

of the plane.

Ans. r&& = 4.62 m/s2, v = 960 km/h


2-18

5: ([1] Problem 2/153)

At the bottom of a loop in the vertical (r-) plane at an altitude of

400 m, the airplane P has a horizontal velocity of 600 km/h and no

horizontal acceleration. The radius of curvature of the loop is 1,200 m.

For the radar tracking at O, determine the recorded values of r&& and &&

for this instant.

Ans. r&& = 12.15 m/s2, && = 0.0365 rad/s2

Chapter 2 Kinematics of Particles Relative Motion (Translating Axes)

2-19

.K (Relative Motion)

1: ([1] Problem 2/189)

Rapid-transit trains A and B travel on parallel tracks. Train A has a speed of 80 km/h and is slowing at

the rate of 2 m/s2, while train B has a constant speed of 40 km/h. Determine the velocity and acceleration of

train B relative to train A.

Ans. /B Avv

= 120 iv

km/h, /B Aav

= -2 iv

m/s2


2-20

2: ([1] Problem 2/192)

Train A travels with a constant speed vA = 120 km/h

along the straight and level track. The driver of car B,

anticipating the railway grade crossing C, decreases the car

speed of 90 km/h at the rate of 3 m/s2. Determine the

velocity and acceleration of the train relative to the car.

Ans. /A Bvv

= 85 km/h (-33.5o), /A Bav

= 3 m/s2 (60o)


2-21

3: ([1] Problem 2/203)

Car A is traveling at the constant speed of 60 km/h as it

rounds the circular curve of 300-m radius and at the instant

represented is at the position = 45o. Car B is traveling at the

constant speed of 80 km/h and passes the center of the circle

at this same instant. Car A is located with respect to car B by

polar coordinates r and with the pole moving with B. For

this instant determine /A Bv and the values of r& and & as

measured by an observer in car B.

Ans. /A Bv = 36.0 m/s

r& = -15.71 m/s, & = 0.1079 rad/s


2-22

4: ([1] Problem 2/205)

The captain of the small ship capable of making a

speed of 6 knots through still water desires to set a course

which will take the boat due east from A to B a distance of

10 nautical miles. To allow for a steady 2-knot current

running northeast, determine his necessary compass

heading H, measured clockwise from the north to the

nearest degree. Also determine the time t of the trip.

(Recall that 1 knot is 1 nautical mile per hour.)

Ans. H = 104o, t = 1 h 23 min


2-23

5: ([2] Problem 12-205)

A man can row a boat at 5 m/s in still water. He wishes to cross a 50-m-

wide river to point B, 50 m downstream. If the river flows with a velocity of 2

m/s, determine the speed of the boat and the time needed to make the crossing.

Ans. v = 6.21 m/s, t = 11.39 min


2-24

6: ([1] Problem 2/211)

A batter hits the baseball A with an initial velocity of v0 = 30 m/s directly toward fielder B at an angle of

30o to the horizontal; the initial position of the ball is 0.9 m above ground level. Fielder B requires 14

s to

judge where the ball should be caught and beings moving to that position with constant speed. Because of

great experience, fielder B chooses his running speed so that he arrives at the catch position

simultaneously with the baseball. The catch position is the field location at which the ball altitude is 2.1 m.

Determine the velocity of the ball relative to the fielder at the instant the catch is made.

Ans. /A Bvv

= 21.5 iv

- 14.19 jv

m/s

Chapter 2 Kinematics of Particles Constrained Motion of Connected Particles

2-25

.P G (Constrained Motion of Particles)

1: ([1] Problem 2/213)

If block A has a velocity of 0.6 m/s to the right, determine

the velocity of cylinder B.

Ans. Bvv

= 1.8 m/s down


2-26

2: ([1] Problem 2/215)

At a certain instant, cylinder A has a downward velocity of 0.8 m/s

and an upward acceleration of 2 m/s2. Determine the corresponding

velocity and acceleration of cylinder B.

Ans. Bvv

= 1.2 m/s up, Bvv

= 3 m/s2 down


2-27

3: ([1] Problem 2/217)

A truck equipped with a power winch on its front end pulls

itself up a steep incline with the cable and pulley arrangement

shown. If the cable is wound up on the drum at the constant rate

of 40 mm/s, how long does it take for the truck to move 4 m up

the incline?

Ans. t = 3 min 20 s


2-28

4: ([1] Problem 2/218)

For the pulley system shown, each of the cables at A and B is

given a velocity of 2 m/s in the direction of the arrow. Determine

the upward velocity v of the load m.

Ans. mvv

= 1.5 m/s up


2-29

5: ([2] Problem 12-183.)

If the end of the cable at A is pulled down with a speed of 2 m/s,

determine the speed at which block E rises.

Ans. cvv

= 0.250 m/s up


2-30

6: ([1] Problem 2/224)

Determine the vertical rise h of the load W during 10 seconds if the

hoisting drum draws in cable at the constant rate of 180 mm/s.

Ans. h = 0.3 m


2-31

7: ([2] Problem 12-174.)

Determine the constant speed at which the cable at A must be drawn

in by the motor in order to hoist the load at B 4.5 m in 5 s.

Ans. Av = 3.6 m/s


2-32

8: ([1] Problem 2/225)

The power winches on the industrial scaffold enable it to be raised or lowered. For rotation in the sense

indicated, the scaffold is being raised. If each drum has a diameter of 200 mm and turns at the rate of 40

rev/min, determine the upward velocity v of the scaffold.

Ans. v = 83.8 mm/s


2-33

9: ([1] Problem 2/226)

The scaffold of Prob. 2/225 is modified here by placing the power winches on the ground instead of on

the scaffold. Other conditions remain the same. Calculate the upward velocity v of the scaffold.

Ans. v = 104.7 mm/s


2-34

10: ([2] Problem 2/12-191.)

The man pulls the boy up to the tree limb C by walking backward. If he starts from rest when xA = 0 and

moves backward with a constant acceleration aA = 0.2 m/s2, determine the speed of the boy at the instant

yB= 4 m. Neglect the size of the limb. When xA= 0, yB = 8 m, so that A and B are coincident, i.e., the rope is

16 m long.

Ans. v = 1.41 m/s up

3

[Kinetics of Particles]

Chapter 3 Kinetics of Particles Section A: Force, Mass, and Acceleration

3A-1

A. (Rectilinear Motion)

1: ([1] Problem 3/1)

During a brake test, the rear-engine car is stopped from an initial speed of 100 km/h in a distance of 50

m. If it is known that all four wheels contribute equally to the braking force, determine the braking force F

at each wheel. Assume a constant deceleration for the 1,500-kg car.

Ans. F = 2,890 N


3A-2

2: ([2] Problem *13-4.)

The van is traveling at 20 km/h when the coupling of the trailer at A fails. If the trailer has a mass of

250 kg and coasts 45 m before coming to rest, determine the constant horizontal force F created by rolling

friction which causes the trailer to stop.

Ans. F = 85.7 N


3A-3

3: ([2] Problem 13-6.)

The baggage truck A has a mass of 800 kg and is used to pull the two cars, each with mass 300 kg. If

the tractive force F on the truck is F = 480 N, determine the initial acceleration of the truck. What is the

acceleration of the truck if the coupling at C suddenly fails? The car wheels are free to roll. Neglect the mass

of the wheels.

Ans. a = 0.343 m/s2, a = 0.436 m/s2


3A-4

4: ([1] Problem 3/2)

The 50-kg crate is stationary when the force P is applied.

Determine the resulting acceleration of the crate if (a) P = 0,

(b) P = 150 N, and (c) P = 300 N.

Ans. (a) 1.12 m/s2 down, (b) 0, (c) 2.04 m/s2 up


3A-5

5: ([1] Problem 3/8)

The 80-kg man in the bosuns chair exerts a pull of 270 N on the rope

for a short interval. Find his acceleration. Neglect the mass of the chair, rope,

and pulleys.

Ans. a = 0.315 m/s2 up

6: ([1] Problem 3/9)

A man pulls himself up the 15o incline by the method shown. If

the combined mass of the man and cart is 100 kg, determine the

acceleration of the cart if the man exerts a pull of 250 N on the rope.

Neglect all friction and the mass of the rope, pulleys, and wheels.

Ans. a = 4.96 m/s2 up


3A-6

7: ([2] Problem 13-21.)

The winding drum D is drawing in the cable at an acceleration

rate of 5 m/s2. Determine the cable tension if the suspended crate

has a mass of 800 kg.

Ans. T = 4.92 kN


3A-7

8: ([1] Problem 3/16)

A small package is deposited by the conveyor belt onto

the 30o ramp at A with a velocity of 0.8 m/s. Calculate the

distance s on the level surface BC at which the package comes

to rest. The coefficient of kinetics friction from A to C is 0.3.

Ans. s = 1.71 m


3A-8

9: ([1] Problem 3/36)

The nonlinear spring has a tensile force-deflection

relationship given by Fs = 150x + 400x2, where x is in meters and

Fs is in newtons. Determine the acceleration of the 6-kg block if

it is released from rest at (a) x = 50mm and (b) x = 100 mm.

Ans. (a) 0 m/s2, (b) 0.7142 m/s2


3A-9

10: ([1] Problem 3/39)

Compute the acceleration of block A for the instant

depicted. Neglect the masses of the pulleys.

Ans. a = 1.406 m/s2


3A-10

: ([1] Problem 3/45)

The sliders A and B are connected by a light rigid bar and move with negligible friction in the slots,

both of which lie in a horizontal plane. For the position shown, the velocity of A is 0.4 m/s to the right.

Determine the acceleration of each slider and the force in the bar at this instant.

Ans. aA = 7.95 m/s2, aB = 8.04 m/s

2, T = 25.0 N


3A-11

A.2 (Curvilinear Motion)

: ([1] Problem 3/51)

If the 2-kg block passes over the top B of the circular

portion of the path with a speed of 3.5 m/s, calculate the

magnitude NB of the normal force exerted by the path on the

block. Determine the maximum speed v which the block can

have at A without losing contact with the path.

Ans. NB = 9.41 N, v = 4.52 m/s


3A-12

K: ([1] Problem 3/53)

If the 80-kg ski-jumper attains a speed of 25 m/s

as he approaches the takeoff position, calculate the

magnitude N of the normal force exerted by the snow

on his skis just before he reaches A.

Ans. N = 1,791 N


3A-13

: ([1] Problem 3/58)

The member OA rotates about a horizontal axis through O

with a constant counterclockwise velocity = 3 rad/s. As it passes

the position = 0, a small block of mass m is placed on it at a

radial distance r = 450 mm. If the block is observed to slip at =

50o, determine the coefficient of static friction s between the

block and the member.

Ans. s = 0.55


3A-14

L: ([1] Problem 3/M)

As the skateboarder negotiates the surface shown,

his mass-center speeds at = 0+, 45o, and 90o are 8.5

m/s, 6 m/s, and 0, respectively. Determine the normal

force between the surface and the skateboard wheels if

the combine mass of the person and skateboard is 70 kg

and his center of mass is 750 mm from the surface.

Ans. N0 = 2,040 N, N45o = 1,158 N, N90o = 0


3A-15

N: ([2] Problem 13-MO.)

Determine the constant speed of the passengers

on the amusement-park ride if it is observed that the

supporting cables are directed at = 30o from the

vertical. Each chair including its passenger has a

mass of 80 kg. Also, what are the components of

force in the n, t, and b directions which the chair

exerts on a 50-kg passenger during the motion?

Ans. v = 6.30 m/s, Fn = 283 N, Ft = 0, Fb = 490 N


3A-16

M: ([1] Problem 3/ML)

Calculate the necessary rotational speed N for the aerial ride in

an amusement park in the order that the arms of gondolas will assume

an angle = 60o with the vertical. Neglect the mass of the arms to

which the gondolas are attached and treat each gondola as a particle.

Ans. N = 11.53 rpm


3A-17

O: ([1] Problem 3/81)

Beginning from rest when = 20o, a 35-kg child

slides with negligible friction down the sliding board

which is in the shape of a 2.5-m circular are. Determine

the tangential acceleration and speed of the child, and the

normal force exerted on her (a) when = 30o and (b)

when = 90o

Ans. (a) at = 8.50 m/s2, v = 2.78 m/s, N = 280 N

(b) at = 0, v = 5.68 m/s, N = 795 N


3A-18

P: ([1] Problem 3/91)

The 1,500-kg car is traveling at 100 km/h on the straight

portion of the road, and then its speed is reduced uniformly

from A to C, at which point it comes to rest. Compute the

magnitude F of the total friction force exerted by the road on

the car (a) just before it passes point B, (b) just after it passes

point B and (c) just before it stops at point C.

Ans. (a) F = 7.83 kN, (b) F = 11.34 kN, (c) F = 7.83 kN


3A-19

9: ([1] Problem 3/82)

Determine the speed v at which the race car will have no

tendency to slip sideways on the banked track, that is, the speed at

which there is on reliance on friction. In addition, determine the

minimum and maximum speeds, using the coefficient of static

friction s = 0.90. Sate any assumptions.

Ans. For no slipping tendency; v = 160 km/h For using s; vmax = 370 km/h, vmin = 0

Chapter 3 Kinetics of Particles Section B: Work and Energy

3B-1

1: ([1] Problem 3/103)

Use the work-energy method to develop an expression for the maximum

height attained by a projectile which is launched with initial speed v0 from

ground level. Evaluate your expression for v0 = 50 m/s. Assume a constant

gravitational acceleration and neglect air resistance.

Ans. 20

2

vh

g= , h = 127.4 m

2: ([1] Problem 3/104)

The spring is unstretched at the position x = 0. Under the

action of a force P, the cart moves from the initial position x1 =

-150 mm to the final position x2 = 80 mm. Determine (a) the

work done on the cart by the spring and (b) the work done on

the cart by its weight.

Ans. springU = 4.03 J, weightU = -3.50 J


3B-2

3: ([1] Problem 3/105)

The small cart has a speed vA = 4 m/s as it passes point

A. It moves without appreciable friction and passes over the

top hump of the track. Determine the cart speed as it passes

point B. Is knowledge of the shape of the track necessary?

Ans. Bv = 7.16 m/s

4: ([1] Problem 3/107)

The 0.5-kg collar C starts from rest at A and slides with negligible

friction on the fixed rod in the vertical plane. Determine the velocity v

with which the collar strikes end B when acted upon by the 5-N force,

which is constant in direction. Neglect the small dimension of the collar.

Ans. v = 2.32 m/s


3B-3

5: ([1] Problem 3/110)

The 15-kg collar A is released from rest in the position shown and

slides with negligible friction up the fixed rod inclined 30o from the

horizontal under the action of a constant force P = 200 N applied to the

cable. Calculate the required stiffness k of the spring so that its maximum

deflection equals 180 mm. The position of the small pulley at B is fixed.

Ans. k = 1,957.41 N/m

6: ([1] Problem 3/111)

In the design of a spring bumper for a 1,500-kg car, it is desired to

bring the car to a stop from a speed of 8 km/h in a distance equal to 150

mm of spring deformation. Specify the required stiffness k for each of

the two springs behind the bumper. The springs are undeformed at the

start of impact.

Ans. k = 164.6 kN/m


3B-4

7: ([1] Problem 3/114)

The man and his bicycle have a combined mass of 95 kg. What

power P is the man developing in riding up a 5-percent grade at a

constant speed of 20 km/h?

Ans. P = 258.515 W

8: ([1] Problem 3/115)

In the design of a conveyor-belt system, small metal blocks are discharged with a velocity of 0.4 m/s

onto a ramp by the upper conveyor belt shown. If the coefficient of kinetics friction between the blocks and

the ramp is 0.30, calculate the angle which the ramp must make with the horizontal so that the blocks will

transfer without slipping to the lower conveyor belt moving at the speed of 0.14 m/s.

Ans. = 16.62o


3B-5

9: ([2] Problem 14-49.)

The 250-N ( 25-kg) crate is given a speed of 3 m/s in t = 4 s

starting from rest. If the acceleration is constant, determine the

power that must be supplied to the motor when t = 2 s. The motor

has an efficiency e = 0.76. Neglect the mass of the pulley and cable.

Ans. inP = 531.2 W


3B-6

10: ([1] Problem 3/133)

Once under way at a steady speed, the 1,000-kg elevator A rises at the rate of

1 story (3 m) per second. Determine the power input Pin into the motor unit M if

the combined mechanical and electrical efficient of the system is e = 0.8.

Ans. inP = 36.8 kW


3B-7

11: ([1] Problem 3/123)

The motor unit A is used to elevate the 300-kg cylinder at a

constant rate of 2 m/s. If the power meter B registers an electrical

input of 2.20 kW, calculate the combine electrical and mechanical

efficient e of the system

Ans. e = 0.892


3B-8

12: ([2] Problem 14-55.)

The elevator E and its freight have a total mass of 400 kg.

Hoisting is provided by the motor M and the 60-kg block C. If the

motor has an efficient of e = 0.6, determine the power that must be

supplied to the motor when the elevator is hoisted upward at a

constant speed of vE = 4 m/s.

Ans. inP = 22.2 kW


3B-9

13: ([1] Problem 3/130)

The 150-kg carriage has an initial velocity of 3 m/s

down the incline at A, when a constant force of 550 N is

applied to the hoisting cable as shown. Calculate the velocity

of the carriage when it reaches B. Show that in the absence

of friction this velocity is independent of whether the initial

velocity of the carriage at A was up or down the incline.

Ans. Bv = 5.51 m/s


3B-10

14: ([1] Problem 3/134)

The system is released from rest in the position shown. The

15-kg cylinder falls through the hole in the support, but the 15-kg

collar (shown in section) is removed from the cylinder as it hits the

support. Determine the distance s which the 50-kg block moves up

the incline. The coefficient of kinetic friction between the block

and the incline is 0.30, and the mass of the pulley is negligible.

Ans. s = 1.67 m


3B-11

15: ([1] Problem 3/143)

The 0.60-kg collar slides on the curved rod in the vertical

plane with negligible friction under the action of a constant force

F in the cord guided by the small pulleys at D. If the collar is

released from rest at A, determine the force F which will result in

the collar striking the stop at B with a velocity of 4 m/s.

Ans. F = 13.21 N

16: ([1] Problem 3/145)

The 10-kg block is released from rest on the horizontal

surface at point B, where the spring has been stretched a distance

of 0.5 m from its neutral position A. The coefficient of kinetics

friction between the block and the plane is 0.30. Calculate (a) the

velocity v of the block as it passes point A and (b) the maximum

distance x to the left of A which the block goes.

Ans. (a) v = 2.13 m/s, (b) x = 0.304 m


3B-12

17: ([1] Problem 3/147)

The spring has an unstretched length of 0.4 m and a stiffness

of 200 N/m. The 3-kg slider and attached spring are released

from rest at A and move in the vertical plane. Calculate the

velocity v of the slider as it reaches B in the absence of friction.

Ans. (a) v = 1.537 m/s

18: ([1] Problem 3/149)

The 1.2-kg slider is released from rest in position A and

slides without friction along the vertical-plane guide shown.

Determine (a) the speed vB of the slider as it passes position B

and (b) the maximum deflection of the spring.

Ans. (a) vB = 9.40 m/s, (b) = 54.2 mm


3B-13

19: ([1] Problem 3/155)

The light rod is pivoted at O and carries the 2- and 4-kg

particles. If the rod is released form rest at = 60o and swings

in the vertical plane, calculate (a) the velocity v of the 2-kg

particle just before it hits the spring in the dashed position and

(b) the maximum compression x of the spring. Assume that x is

small so that the position of the rod when the spring is

compressed is essential horizontal.

Ans. (a) v = 1.162 m/s, (b) x = 12.07 mm


3B-14

20: ([1] Problem 3/159)

The mechanism shown lies in the vertical plane and is released

from rest in the position for which = 60o. In this position the

spring is unstretched. Calculate the velocity of the 5-kg sphere

when = 90o. The mass of links is small and may be neglected.

Ans. (a) v = 1.736 m/s


3B-15

21: ([1] Problem 3/165)

A satellite is put into an elliptical orbit around the earth and has

a velocity vP at the perigee position P. Determine the expression for

the velocity vA at the apogee position A. The radii to A and P are,

respectively, rA and rP. Note that the total energy remains constant.

Ans. 2 2 1 12A PP A

v v gRr r

=

22: ([1] Problem 3/167)

Upon its return voyage from a space mission, the spacecraft

has a velocity of 24,000 km/h at position A, which is 7,000 km

from the center of the earth. Determine the velocity of the

spacecraft when it reaches point B, which is 6,500 km from these

two points is outside the effect of the earths atmosphere.

Ans. Bv = 26,300 km/h


3B-16

23: ([1] Problem 3/168)

The 2-kg sliding collar C with attached spring moves with

friction from A to B along the fixed rod. If the collar has a

velocity of 3 m/s at A and a velocity of 5 m/s at B, determine

the loss Uf of energy due to friction. The spring has a stiffness

of 30 N/m and an unstretched length of 0.5 m. The x-y plane is

horizontal. Also determine the average friction force F during

the motion from A to B.

Ans. fU = -27.65 J, F = 13.825 N

Chapter 3 Kinetics of Particles Section C: Impulse and Momentum

3C-1

C. (Linear Impulse and Linear Momentum)

6 1: ([1] Problem 3/187)

A 75-g projectile traveling at 600 m/s strikes and

becomes embedded in the 50-kg block, which is initially

stationary. Compute the energy lost during the impact.

Express your answer as an absolute value E and as a

percentage n of the original system energy E.

Ans. E = 13,480 J, n = 99.9%


3C-2

6 2: ([1] Problem 3/188)

A 60-g bullet is fired horizontally with a velocity v1

= 600 m/s into the 3-kg block of soft wood initially at

rest on the horizontal surface. The bullet emerges from

the block with the velocity v2 = 400 m/s, and the block

is observed to slide a distance of 2.70 m before coming

to rest. Determine the coefficient of kinetic friction k

between the block and the supporting surface.

Ans. k = 0.302


3C-3

6 3: ([1] Problem 3/190)

A 45-kg boy runs and jumps on his 10-kg sled with a horizontal velocity of 4.6 m/s. If the sled and boy

coast 25 m on the snow before coming to rest, compute the coefficient of kinetic friction between the snow

and the runners of the sled.

Ans. k = 0.029


3C-4

6 4: ([1] Problem 3/192)

The inspection gondola for a cableway is being drawn up the

sloping cable at the speed of 4 m/s. If the control cable at A

suddenly breaks, calculate the time t after the break occurs for the

gondola to reach a speed of 8 m/s down the incline cable. Neglect

friction and treat the gondola as a particle.

Ans. t = 3.18 s


3C-5

6 5: ([1] Problem 3/193)

The 200-kg lunar lander is descending onto the moons surface with a velocity of 6 m/s

when its retro-engine is fired. If the engine produces a thrust T for 4 s which varies with the

time as shown and then cuts off, calculate the velocity of the lander when t = 5 s, assuming that

it has not yet landed. Gravitational acceleration at the moons surface is 1.62 m/s2.

Ans. v = 2.10 m/s


3C-6

6 6: ([1] Problem 3/195)

The 9-kg block is moving to the right with a

velocity of 0.6 m/s on the horizontal surface when a

force P is applied to it at time t = 0 as shown in the

graph. Calculate the velocity v of the block when t =

0.4 s. The coefficient of kinetics friction is k = 0.3.

Ans. v = 1.823 m/s


3C-7

6 7: ([2] Problem 15-31.)

The log has a mass of 500 kg and rests on the

ground for which the coefficient of static and kinetic

friction are s = 0.5 and k = 0.4, respectively. The

winch delivers a horizontal towing force T to its cable at

A which varies as shown in the graph. Determine the

speed of the log when t = 5 s. Originally the tension in

the cable is zero. Hint: First determine the force needed

to being moving the log.

Ans. v = 7.65 m/s


3C-8

6 8: ([1] Problem 3/207)

Car B is initially stationary and is struck by car A moving with initial speed v1 = 30 km/h. The cars

become entangled and move together with speed v after the collision. If the time duration of the collision is

0.1 s, determine (a) the common final speed v, (b) the average acceleration of each car during the collision,

and (c) the magnitude R of the average force exerted by each car on the other car during the impact. All

brakes are released during the collision.

Ans. (a) v = 20 km/h

(b) Aa = -27.8 m/s2, Ba = 55.6 m/s

2

(c) R = 50 kN


3C-9

6 9: ([1] Problem 3/211)

The ice-hockey puck with a mass of 0.20 kg has a velocity of 12 m/s before being struck by the hockey

stick. After the impact the puck moves in the new direction shown with a velocity of 18 m/s. If the stick is in

contact with the puck for 0.04 s, compute the magnitude of the average force Fv

exerted by the stick on the

puck during contact, and find the angle made by Fv

with the x-direction.

Ans. Fv

= 147.8 N, = 12.02o


3C-10

6 10: ([1] Problem 3/218)

The ballistic pendulum is a simple device to measure projectile

velocity v by observing the maximum angle to which the box of sand

with embedded projectile swings. Calculate the angle if the 60-g

projectile is fired horizontally into the suspended 20-kg box of sand with a

velocity v = 600 m/s. Also find the percentage of energy lost during impact.

Ans. = 23.38o, n = 99.7%


3C-11

6 11: ([1] Problem 3/220)

The cylindrical plug A of mass mA is released from rest at B

and slides down the smooth circular guide. The plug strikes the

block C and becomes embedded in it. Write the expression for

the distance s which the block and plug slide before coming to

rest. The coefficient of kinetic friction between the block and

the horizontal surface is k.

Ans. 2

A

k A C

mrs

m m

= +


3C-12

6 12: ([1] Problem 3/221)

The baseball is traveling with a horizontal velocity of 135 km/h just before impact with the bat. Just

after the impact, the velocity of the 146-g ball is 210 km/h directed at 35o to the horizontal as shown.

Determine the x- and y-components of the average force R exerted by the bat on the baseball during the 0.005-s impact. Comment on the treatment of the weight of the baseball (a) during the impact and (b) over

the first few second after impact.

Ans. Rx = 2490 N, Ry = 978 N


3C-13

C.I (Angular Impulse and Angular Momentum)

6 1: ([1] Problem 3/230)

The small spheres, which have the masses and initial

velocities shown in the figure, strike and become attached to the

spiked ends of the rod, which is freely pivoted at O and is initially

at rest. Determine the angular velocity of the assembly after

impact. Neglect the mass of the rod.

Ans. ( )5 .3

vCCW

L =


3C-14

6 2: ([1] Problem 3/231)

A particle of mass m moves with negligible friction on a

horizontal surface and is connected to a light spring fastened at O. At

position A the particle has the velocity vA = 4 m/s. Determine the

velocity vB of the particle as it passes position B.

Ans. vB = 5.43 m/s


3C-15

6 3: ([1] Problem 3/233)

The assembly starts from rest and reaches an angular speed of

150 rev/min under the action of a 20-N force T applied to the string

for t seconds. Determine the t. Neglect friction and all masses except

those of the four 3-kg spheres, which may be treated as particles.

Ans. t = 15.08 s


3C-16

6 4: ([1] Problem 3/250)

The assembly of two 5-kg spheres is rotation freely about

the vertical axis at 40 rev/min with = 90o. If the force F

which maintains the given position is increased to raise the

base collar and reduce to 60o, determine the new angular

velocity . Also determine the work U done by F in changing

the configuration of the system. Assume that mass of the arms

and collar is negligible.

Ans. = 3.00 rad/s, U = 5.34 J


3C-17

6 5: ([1] Problem 3/236)

The 6-kg spheres and 4-kg block (shown in section) are

secured to the arm of negligible mass which rotate in the

vertical plane about a vertical axis at O. The 2-kg plug is

released from rest at A and falls into the recess in the block

when the arm has reached the horizontal position. An instant

before engagement, the arm has an angular velocity 0 = 2

rad/s. Determine the angular velocity of the arm

immediately after the plug has wedged itself in the block.

Ans. = 0.172 rad/s (CW)


3C-18

6 6: ([2] Problem 15-108.)

A child having a mass of 50 kg holds her legs up as shown

as she swings downward from rest at 1 = 30o. Her center of

mass is located at point G1. When she is at the bottom position

= 0o, she suddenly lets her legs come down, shifting her center of

mass to position G2. Determine her speed upswing due to this

sudden movement and the angle 2 to which she swing before

momentarily coming to rest. Treat the childs body as a particle.

Ans. v = 2.53 m/s, 2 = 27.0o


3C-19

C. (Impact)

6 1: ([2] Problem 15-57.)

Disk A has a mass of 2 kg and is sliding forward on the smooth surface with a velocity (vA)1 = 5 m/s

when is strikes the 4-kg disk B, which is sliding towards A at (vB)1 = 2 m/s, with direction central impact. If

the coefficient of restitution between the disk is e = 0.4, compute the velocity of A and B just after collision.

Ans. (vA)2 = 1.53 m/s , (vB)2 = 1.27 m/s


3C-20

6 2: ([1] Problem 3/251)

As a check of the basketball before the start of a game, the referee

releases the ball from the overhead position shown, and the ball

rebounds to about waist level. Determine the coefficient of restitution e

and the percentage n of the original energy lost during the impact.

Ans. e = 0.724, n = 47.6%


3C-21

6 3: ([1] Problem 3/257)

If the center of the ping-pong ball is to clear the net as shown, at what height h should the ball be

horizontally served? Also determine h2. The coefficient of restitution for the impacts between ball and table

is e = 0.9, and the radius of the ball is r = 18.75 mm.

Ans. h = 273 mm, h2 = 185.8 mm


3C-22

6 4: ([2] Problem 15-67.)

The 2-kg ball is thrown at the suspended 20-kg block with a

velocity of 4 m/s. If the coefficient of restitution between the ball

and the block is e = 0.8. (a) Determine the maximum height h to

which the block will swing before it momentarily stops. (b) If the

time of impact is 0.005 s, determine the average normal force F

exerted on the block during this time.

Ans. (a) h = 21.8 mm

(b) F = 2.62 kN


3C-23

6 5: ([1] Problem 3/263)

The figure shows n spheres of equal mass m suspended in a

line by wires of equal length so that the spheres are almost

touching each other. If sphere 1 is released from the dashed

position and strikes sphere 2 with a velocity v1, write an

expression for the velocity vn of the nth sphere immediately

after being struck by the one adjacent to it. The common

coefficient of restitution is e.

Ans. 1

1

1

2

n

n

ev v

+ =


3C-24

6 6: ([1] Problem 3/275)

The 3000-kg anvil A of the drop forge is mounted on a nest of heavy coil

springs having a combined stiffness of 2.8(10)6 N/m. The 600-kg hammer B

falls 500 mm from rest and strikes the anvil, which suffers a maximum

downward deflection of 24 mm from its equilibrium position. Determine the

height h of rebound of the hammer and the coefficient of restitution e which

applies.

Ans. h = 14.53 mm, e = 0.405


3C-25

6 7: ([1] Problem 3/260)

The steel ball strikes the heavy steel plate with a velocity v0 = 24

m/s at an angle of 60o with the horizontal. If the coefficient of restitution

is e = 0.8, compute the velocity v and its direction with which the ball

rebounds from the plate.

Ans. v = 20.5 m/s, = 54.18o


3C-26

6 8: ([1] Problem 3/261)

The previous problem is modified in that the plate struck by the

ball now has a mass equal to that of the ball and is supported as

shown. Compute the final velocity of the both masses immediately

after impact if the plate is initially stationary and all other condition

are the same as stated in the previous problem.

Ans. Ball v1 = 12.20 m/s, = -9.83o

Plate v2 = 18.71 m/s down


3C-27

6 9: ([1] Problem 3/262)

In a pool game the cue ball A must strike the eight ball in the

position shown in order to send it to the pocket P with a velocity

v2. The cue ball has a velocity v1 before impact and a velocity v1

after impact. The coefficient of restitution is 0.9. Both balls have

the same mass and diameter. Calculate the rebound angle and the

fraction n of the kinetic energy which is lost during the impact.

Ans. = 2.862o, n = 4.7%


3C-28

6 10: ([2] Problem 15-77.)

The cue ball A is given an initial velocity (vA)1 = 5 m/s. If

it makes a direction collision with ball B (e = 0.8), determine

the velocity of B and the angle just after it rebounds from

the cushion at C (e = 0.6). Each ball has a mass of 0.4 kg.

Assume the ball slides without rolling.

Ans. (vB)3 = 3.24 m/s, = 43.9o


3C-29

6 11: ([1] Problem 3/259)

To pass inspection, steel balls designed for use in ball

bearing must clear the fixed bar A at the top of their rebound

when dropped from rest through the vertical distance H = 900

mm onto the heavy inclined steel plate. If balls which have a

coefficient of restitution of less than 0.7 with the rebound

plate are to be rejected, determine the location of the bar by

specifying h and s. Neglect any friction during impact.

Ans. h = 379 mm, s = 339 mm


3C-30

6 12: ([1] Problem 3/276)

A child throws a ball from point A with a speed

of 15 m/s. It strikes the wall at point B and then

returns exactly to point A. Determine the necessary

angle if the coefficient of restitution in the wall

impact is e = 0.5.

Ans. = 11.55o or 78.4o

4

[Kinetics of Systems of Particles]

Chapter 4 Kinetics of Systems of Particles Introduction

4-1

1: ([1] Problem %/1)

The system of three particles has the indicated

particle masses, velocity, and external forces. Determine

rv

, rv& , r

v&& , T, OH

v, and OH

v& for this system.

Ans. ( )4 67

di j kr = + +

vv vv, ( )4 2 6

7i

vr j k= + +

vv vv&

7

Fkr

m=

v&&v

, 213T mv= , ( )12 6 2O i j kH mdv= + +vv vv

O F jH d= v v&

Chapter 4 Kinetics of Systems of Particles Generalized Newtons Second Law

4-2

2: ([1] Problem %/3)

The two 2-kg balls are initially at rest on the horizontal surface

when a vertical force F = 60 N is applied to the junction of the attached

wires as shown. Compute the vertical component ay of the initial

acceleration of each ball by considering the system as a whole.

Ans. ay = 5.19 m/s2

3: ([1] Problem %/4)

Three monkeys A, B, and C with masses of 10, 15, and 8 kg. respectively, are

climbing up and down the rope suspended from D. At the instant represented, A is

descending the rope with an acceleration of 2 m/s2, and C is pulling himself up with an

acceleration of 1.5 m/s2. Monkey B is climbing up with a constant speed of 0.8 m/s. Treat

the rope and monkeys as a complete system and calculate the tension T in the rope at D.

Ans. T = 315.7 N (tension)

Chapter 4 Kinetics of Systems of Particles Generalized Newtons Second Law

4-3

4: ([1] Problem %/6)

The two spheres, each of mass m, and are connected by the

spring and hinged bars of negligible mass. The spheres are free

to slide in the smooth guides up the incline . Determine the

acceleration aC of the center C of the spring.

Ans. aC = sin2

Fg

m

5: ([1] Problem %/7)

Calculate the acceleration of the center of mass of the system of the four

10-kg cylinders. Neglect friction and the mass of the pulleys and cables.

Ans. a = 15.19 m/s2

Chapter 4 Kinetics of Systems of Particles Work and Energy

4-4

6: ([1] Problem %/23)

The three small spheres, each of mass m, are secured to the light

rods to form a rigid unit supported in the vertical plane by the smooth

circular surface. The force of constant magnitude P is applied

perpendicular to one rod at its midpoint. If the unit starts from rest at

= 0, determine (a) the minimum force Pmin which will bring the unit

to rest at = 60o and (b) the common velocity v of spheres 1 and 2

when = 60o if P = 2Pmin.

Ans. (a) Pmin = 9mg

, (b) v = 3 / 2gr

Chapter 4 Kinetics of Systems of Particles Impulse and Momentum

4-5

.: ([1] Problem %/18)

The 300-kg and 400-kg mine cars are rolling in opposite

directions along the horizontal track with the respective speeds of

0.6 m/s and 0.3 m/s. Upon impact the car become coupled together.

Just prior to impact, a 100-kg boulder leaves the delivery chute with

a velocity of 1.2 m/s in the direction shown and lands in the 300-kg

car. Calculate the velocity v of the system after the boulder has

come to rest relative to the car. Would the final velocity be the same

if the cars were coupled before the boulder dropped?

Ans. v = 0.205 m/s

8: ([1] Problem %/19)

The three freight cars are rolling along the horizontal track with the velocities shown. After the

impacts occur, the three cars become coupled together and move with a common velocity v. The

loaded cars A, B, and C have masses of 65 Mg, 50 Mg, and 75 Mg, respectively. Determine v and

calculate the percentage loss n of energy of the system due to coupling.

Ans. v = 0.355 km/h , n = 95%


4-6

9: ([1] Problem %/11)

The two small spheres, each of mass m, are rigidly connected

by a rod of negligible mass. The center C of the rod has a velocity

v in the x-direction, and the rod is rotating counterclockwise at the

constant rate & . For a given value of , write the expressions for

(a) the linear momentum of each sphere and (b) the linear

momentum Gv

of the system of the two spheres.

Ans. (a) ( ) ( )1 sin cosiG b b jm v = + v vv

& &

( ) ( )2 sin cosiG b b jm v = + v vv

& &

(b) 2 iG mv=v v


4-7

10: ([1] Problem %/17)

Two projectiles, each with a mass of 10 kg, are fired

simultaneously from the 1-Mg vehicle shown, which is

moving with an initial velocity v1 = 1.2 m/s in the direction

opposite to the firing. Each of projectile has a muzzle velocity

vr = 1200 m/s relative to the barrel. Calculate the velocity v2

of the vehicle after the projectiles have been fired.

Ans. v2 = 24.7 m/s


4-8

11: ([1] Problem %/15)

The three small spheres are welded to the light rigid frame which

is rotation in a horizontal plane about a vertical axis through O with an

angular velocity & = 20 rad/s. If a couple MO = 30 Nm is applied to

the frame for 5 seconds, compute the new angular velocity & .

Ans. & = 80.7 rad/s


4-9

12: ([1] Problem %/16)

The four 3-kg balls are rigidly mounted to the rotating frame and

shaft, which are initially rotating freely about the vertical z-axis at the

angular rate of 20 rad/s clockwise when viewed from above. If a

constant torque M = 30 Nm is applied to the shaft, calculate the time t

to reverse the direction of rotation and reach an angular velocity & =

20 rad/s in the same sense as M.

Ans. t = 1.36 s, t = 2.72 s


4-10

13: ([1] Problem %/94)

A 60-g bullet is fired horizontally with a velocity v = 300 m/s

into the slender bar of a 1.5-kg pendulum initially at rest. If the bullet

embeds itself in the bar, compute the resulting angular velocity of the

pendulum immediately after the impact. Treat the sphere as a particle

and neglect the mass of the rod. Why is the linear momentum of the

system not conserved?

Ans. = 11.88 rad/s


4-11

14: ([1] Problem %/97)

The two balls are attached to the light rigid rod, which is suspended by

a cord from the support above it. If the balls and rod, initial at rest, are

struck with the force F = 60 N, calculate the corresponding acceleration av

of the mass center and the rate && at which the angular velocity of bar is

changing.

Ans. av

= 20 m/s2, && = 336 rad/s2

Chapter 4 Kinetics of Systems of Particles Work and Energy, Impulse and Momentum

4-12

15: ([1] Problem %/29)

The carriage of mass 2m is free to roll along the

horizontal rails and carries the two spheres, each of

mass m, mounted on rods of length l and negligible

mass. The shaft to which the rods are secured is

mounted in the carriage and is free to rotate. If the

system is released from rest with the rods in the vertical

position where = 0, determine the velocity vx of the

carriage and the angular velocity & of the rods for

instant when = 180o. Treat the carriage and the spheres

as particles and neglect any friction.

Ans. vx = 2gl , 2

2g

l =&

5

[Kinematics of Rigid Bodies]

Chapter 5 Plane Kinematics of Rigid Bodies Rotation

5-1

. (Rotation)

1: ([1] Problem 5/3)

The angular velocity of a gear is controlled according to = 12-3t2 where , in radians per second, is

positive in the clockwise sense and where t is the time in the seconds. Find the net angular displacement

from the time t = 0 to t = 3 s. Also find the total number of revolutions N through which the gear turns

during the 3 seconds.

Ans. = rad, N = rev

2: ([1] Problem 5/5)

Magnetic tape is fed over and around the light pulleys mounted

in a computer frame. If the speed v of the tape is constant and if the

ratio of the magnitudes of the acceleration of points A and B is 2/3,

determine the radius r of the larger pulley.

Ans. r = 112.5 mm


5-2

3: ([1] Problem 5/9)

The circular disk rotates about its center O in the direction shown. At

a certain instant point P on the rim has an acceleration given by

3 4a i j= v vv

m/s2. For this instant determine the angular velocity v

and

angular acceleration v

of the disk.

Ans. 8 k = v v

rad/s, 6 k =v v rad/s2


5-3

4: ([1] Problem 5/12)

The right-angle bar rotates about the z-axis through O with

an angular acceleration = 3 rad/s2 in the direction shown.

Determine the velocity and acceleration of point P when the

angular velocity reaches the value = 2 rad/s.

Ans. 0.4P iv j= +v vv m/s; 1.077Pv =

v m/s

1.4 2.3P i ja = v vv m/s2; 2.69Pa =

v m/s2


5-4

5: ([1] Problem 5/13)

The circular disk rotates with a constant angular velocity = 40 rad/s

about its axis, which is inclined in the y-z plane at the angle = tan-1 34

.

Determine the vector expressions for the velocity and acceleration of point

P, whose position vector at the instant shown is 375 400 300r i j k= + vv vv

mm. (Check the magnitude of your results from the scalar values v = r

and an = r2.)

Ans.

20 12 9P i j kv = + vv vv

m/s; 25Pv =v

m/s

600 640 480P i j ka = +vv vv

m/s2; 1000Pa =v

m/s2


5-5

6: ([1] Problem 5/17)

The two attached pulleys are driven by the belt with increasing speed.

When the belt reaches a speed v = 0.6 m/s, the total acceleration of point P

is 8 m/s2. For this instant determine the angular acceleration of the

pulleys and the acceleration of point B on the belt.

Ans. = 17.44 rad/s2, aB = 1.744 m/s2


5-6

7: ([1] Problem 5/22)

The two V-belt pulleys from an integral unit and rotate about

the fixed axis at O. At a certain instant, point A on the belt of the

smaller pulley has a velocity vA = 1.5 m/s, and point B on the

larger pulley has a acceleration aB = 45 m/s2 as shown. For this

instant determine the magnitude of the acceleration Cav

of point

C and sketch the vector in your solution.

Ans. Cav

= 149.59 m/s2

8: ([1] Problem 5/25)

The circular disk rotates about its center O. At a certain instant point A has a

velocity vA = 0.8 m/s in the direction shown, and at the same instant the tangent

of angle made by the total acceleration vector of any point B with its radial line

to O is 0.6. For this instant compute the angular acceleration of the disk.

Ans. = 38.4 rad/s2


5-7

9: ([1] Problem 5/26)

A V-belt speed-reduction drive is shown where pulley

A drives the two integral pulleys B which in turn drive

pulley C. If A starts from rest at time t = 0 and is given a

constant angular acceleration 1, derive expressions for the

angular velocity of C and the magnitude of the acceleration

of a point P on the belt, both at time t.

Ans. 2

11

2C

rt

r =

422 41 1

1 12 2

1Pr r

a tr r = +

Chapter 5 Plane Kinematics of Rigid Bodies Absolute Motion

5-8

.> (Absolute Motion)

1: ([1] Problem 5/27)

Slider A moves in the horizontal slot with a constant speed v for a

short interval of motion. Determine the angular velocity of bar AB in

terms of the displacement xA.

Ans. 2

11

2C

rt

r =


5-9

2: ([1] Problem 5/29)

Point A is given a constant acceleration a to the right starting from rest with x essentially zero.

Determine the angular velocity of link AB in terms of x and a.

Ans. 2 2

2

4

ax

b x =


5-10

3: ([1] Problem 5/37)

Calculate the angular velocity of the slender bar AB

as a function of the distance x and the constant angular

velocity 0 of the drum.

Ans. 02 2rh

x h

=

+


5-11

4: ([1] Problem 5/44)

Derive an expression for the upward velocity v of the car

hoist in terms of . The piston rod of the hydraulic cylinder is

extending at the rate s& .

Ans. 2 22cot 2 cosv s L b bLL = + &


5-12

5: ([1] Problem 5/46)

Determine the acceleration of the shaft B for = 60o if the crank OA

has an angular acceleration && = 8 rad/s2 and angular velocity & = 4 rad/s at

this position. The spring maintains contact between the roller and the

surface of the plunger.

Ans. aB = 788.51 mm/s2


5-13

6: ([1] Problem 5/48)

The hydraulic cylinder C gives end A of link AB and a

constant velocity v0 in the negative x-direction. Determine

expression for the angular velocity = & and angular acceleration = && of the link in terms of x.

Ans. 20

2 2

v

L x = =

& ,

( )

20

3 22 2

xv

L x

= =

&&


5-14

7: ([2] Problem 16-38.)

The crankshaft AB is rotating at a constant

angular velocity = 150 rad/s. Determine the velocity of the piston P at the instant = 30o

Ans. vP = 18.5 m/s


5-15

8: ([1] Problem 5/57)

The punch is operated by a simple harmonic oscillation of the pivoted sector

given by given by = 0sin2t, where the amplitude is 0 = /12 rad. (15o) and the

time for one complete oscillation is 1 second. Determine the acceleration of the

punch when (a) = 0 and (b) = /12.

Ans. (a) a = 0.909 m/s2 up, (b) a = 0.918 m/s2 down


5-16

9: ([1] Problem 5/31)

The concreate pier P is being lowered by the pulley and cable

arrangement shown. If points A and B have velocities of 0.4 m/s and 0.2 m/s,

respectively, compute the velocity of P, the velocity of point C for instant

represented, and the angular velocity of the pulley.

Ans. = 0.5 rad/s CW, vP = 0.3 m/s, vC = 0.25 m/s


5-17

10: ([1] Problem 5/34)

The spool rolls on its hub up the inner cable A as the equalizer

plate B pulls the outer cables down. The three cables are wrapped

securely around their respective peripheries and do not slip. If, at the

instant represented, B has moved down a distance of 1600 mm from

rest with a constant acceleration of 0.2 m/s2, determine the velocity of

point C and the acceleration of the center O for this particular instant.

Ans. vC = 1.2 m/s , aO = 0.05 m/s2

Chapter 5 Plane Kinematics of Rigid Bodies Relative Velocity

5-18

.3 (Relative Velocity)

: ([1] Problem 5/L)

End A of the link has the velocity shown at the instant

depicted. End B is confined to move in the slot. For this instant

calculate the velocity of B and the angular velocity of AB.

Ans. vB = 3.06 m/s, AB = 7.88 rad/s CCW


5-19

2: ([1] Problem 5/63)

For a short interval, collars A and B are sliding along the fixed vertical

shaft with velocities vA = 2 m/s and vB = 3 m/s in the directions shown.

Determine the magnitude of the velocity of point C for the position = 60o.

Ans. vC = 1.528 m/s


5-20

3: ([1] Problem 5/69)

The rider of the bicycle shown pumps steadily to maintain a

constant speed of 16 km/h against a slight head wind. Calculate the

maximum and minimum magnitude of the absolute velocity of the

pedal A.

Ans. (vA)max = 5.33 m/s, (vA)min = 3.56 m/s


5-21

4: ([2] Problem *16-60.)

The rotation of link AB creates an oscillating

movement of gear F. If AB has an angular velocity AB

= 6 rad/s, determine the angular velocity of gear F at the

instant shown. Gear E is rigidly attach to arm CD and

pinned at D to a fixed point.

Ans. F = 12.0 rad/s

Chapter 5 Plane Kinematics of Rigid Bodies Instantaneous Center of Zero Velocity

5-22

.4 (Instantaneous Center of Zero Velocity)

1: ([1] Problem 5/94)

End A of the link has a downward velocity vA of 2 m/s

during an interval of its motion. For the position where = 30o, determine by the method of this article the angular velocity of AB and the velocity vG of the midpoint G of the link.

Ans. AB = 11.547 rad/s CW, vG = 1.1547 m/s

2: ([1] Problem 5/96)

For the instant represented, when crank OA passes

the horizontal position, determine the velocity of the

center G of link AB by the method of this article.

Ans. AB = 11.547 rad/s CW, vG = 1.1547 m/s


5-23

3: ([1] Problem 5/109)

Horizontal oscillation of the spring-loaded plunger

E is controlled by varying the air pressure in the

horizontal pneumatic cylinder F. If the plunger has a

velocity of 2 m/s to the right when = 30o, determine

the downward velocity vD of roller D in the vertical

guide and find the angular velocity of ABD for this

position.

Ans. vD = 2.31 m/s, ABD = 13.33 rad/s CW

02-ENGINEERING DYNAMICS (Worksheets) - (V 2558)

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Transcript of 02-ENGINEERING DYNAMICS (Worksheets) - (V 2558)