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Unformatted text preview: PROBLEMS 13.155 The coefficient of restitution between the two collars is known to
be 0.80. Determine ((1) their velocities after impact, (19) the energy
loss during impact. 13.156 Collars A and B, of the same mass m, are moving toward each
other with the velocities shown. Knowing that the coefficient of
restitution between the collars is 0 (plastic impact), show that after
impact (a) the common velocity of the collars is equal to half the
difference in their speed before impact, (b) the loss in kinetic energy is %m(vA + 03)2. 13.157 TWO steel blocks are sliding on a frictionless horizontal surface
with the velocities shown. Knowing that after impact the velocity
of B is observed to be 10.5 ft/s to the right, determine the coeffi
cient of restitution between the two blocks. 13.158 Two steel blocks are sliding on a frictionless horizontal surface
with the velocities shown. Knowing that the coefficient of restitu—
tion between the two blocks is 0.75, determine (a) the velocity of
each block after impact, (b) the loss of kinetic energy due to the
impact. 13.159 Two identical cars A and B are at rest on a loading dock with brakes
released. Car C, of a slightly different style but of the same weight,
has been pushed by dockworkers and hits car B with a velocity of
1.5 m/s. Knowing that the coefficient of restitution is 0.8 between
B and C and 0.5 between A and B, determine the velocity of each
car after all collisions have taken place. Fig. P13.159 13.160 Three steel spheres of equal weight are suspended from the ceiling
by cords of equal length which are spaced at a distance slightly
greater than the diameter of the spheres. After being pulled back
and released, sphere A hits sphere B, which then hits sphere C.
Denoting by e the coefficient of restitution between the spheres and
by v0 the velocity of A just before it hits B, determine (a) the veloci
ties of A and B immediately after the ﬁrst collision, (b) the velocities
of B and C immediately after the second collision. (0) Assuming now
that n spheres are suspended from the ceiling and that the ﬁrst
sphere is pulled back and released as described above, determine
the velocity of the last sphere after it is hit for the ﬁrst time. (d) Use
the result of part c to obtain the velocity of the last sphere when
n = 6 ande = 0.95. ﬂ“ WW 5 kg
Fig. P13.155 Fig. P13.157 and "3.158 Fig. “3.160 .fficialr
rom a.
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act is when
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pact. 13.166 Two identical hockey pucks are moving on a hockey rink at the
same speed of 3 m/s and in parallel and opposite directions when
they strike each other as shown. Assuming a coefficient of restitu—
tion e = 1, determine the magnitude and direction of the velocity of each puck after impact. 13.167 Two identical pool balls of 2.37in.diameter, may move freely on
a pool table. Ball B is at rest and ball A has an initial velocity v = voi. ((1) Knowing that b = 2 in. and e = 0.7, determine the
velocity of each ball after impact. (b) Show that if e = 1, the final
velocities of the balls form a right angle for all values of b. Fig. "3.167 13.168 The coefficient of restitution is 0.9 between the two 2.37in.
diameter billiard balls A and B. Ball A is moving in the direction
shown with a velocity of 3 ft/s when it strikes ball B, which is at
rest. Knowing that after impact B is moving in the x direction,
determine (a) the angle 9, (b) the velocity of B after impact. Fig. pmas 13.169 A boy located at point A halfway between the center 0 of a semi
circular wall and the wall itself throws a ball at the wall in a direc—
tion forming an angle of 45° with CA. Knowing that after hitting
the wall the ball rebounds in a direction parallel to OA, determine
the coefficient of restitution between the ball and the wall. Fig. “3.166 Problems 837 Fig. “3.169 i 13.174 tude
the
dis
ball 5 13.175 13.176 ngle
ball >oint _d d, A lkg block B is moving with a velocity v0 of magnitude 00 = 2 m/s
as it hits the 0.5kg sphere A, which is at rest and hanging from a
cord attached at 0. Knowing that pk = 0.6 between the block and
the horizontal surface and e = 0.8 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. A 1.5—kg block B is attached to an undeformed spring of constant
k = 80 N/m and is resting on a horizontal frictionless surface when
it is struck by an identical block A moving at a speed of 5 m/s. Con
sidering successively the cases when the coefficient of restitution
between the two blocks is (1) e = 1, (2) e = 0, determine (a) the
maximum deflection of the spring, ([9) the ﬁnal velocity of block A. Block A is released from rest and slides down the frictionless sur
face of B until it hits a bumper on the right end of B. Block A has
a mass of 10 kg and object B has a mass of 30 kg and B can roll
freely on the ground. Determine the velocities of A and B imme
diately after impact when (a) e = 0, (b) e = 0.7. A Fig. P13.'l76 13.177 A QOg ball thrown with a horizontal velocity v0 strikes a 720g plate ' 30° attached to a vertical wall at a height of 900 mm above the ground.
It is observed that after rebounding, the ball hits the ground at a
distance of 480 mm from the wall when the plate is rigidly attached
to the wall (Fig. l) and at a distance of 220 mm when a foamrubber
mat is placed between the plate and the wall (Fig. 2). Determine
(a) the coefﬁcient of restitution e between the ball and the plate,
(I?) the initial velocity v.) of the ball. Fig. P13.'l77 Fig. P13.I75 Problems 839 —1b. 13.181 Blocks A and B each weigh 0.8 lb and block C weighs 2.4 lb. The Pmb'ems 84] llly coefficient of friction between the blocks and the plane is m. = 0.30.
:he Initially block A is moving at a speed 00 = 15 ft/s and blocks B
led and C are at rest (Fig. 1). After A strikes B and B strikes C, all
ate mine (a) the coefficients of restitution between A and B and ing three blocks come to a stop in the positions shown (Fig. 2). Deter— ' ‘ l
between B and C, (b) the displacement x of block C. I Fig. ”3.181 [ 13.182 The three blocks shown are identical. Blocks B and C are at rest '1.b when block B is hit by block A, which is moving with a velocity VA 1 :
n1 of 3 ft/s. After the impact, which is assumed to be perfectly plastic '
) a (e = 0), the velocity of blocks A and B decreases due to friction, _
EOf while block C picks up speed, until all three blocks are moving .. ;
or with the same velocity v. Knowing that the coefficient of kinetic . ' " 'i
Xl' friction between all surfaces is 11;. = 0.20, determine (a) the time   '  ' '
ue . required for the three blocks to reach the same velocity, ([9) the total '
.' distance traveled by each block during that time. '_ '
Se: 13.183 After having been pushed by an airline employee, an empty 40kg . i '
w luggage carrier A hits with a velocity of 5 m/s an identical carrier '3
lat B containing a 15—kg suitcase equipped with rollers. The impact 600
he '._ causes the suitcase to roll into the left wall of carrier B. Knowing mfg 3
m that the coefficient of restitution between the two carriers is 0.80 200 :
en and that the coefficient of restitution between the suitcase and the
wall of carrier B is 0.30, determine (a) the velocity of carrier B ' ,
after the suitcase hits its wall for the first time, (b) the total energy '
lost in that impact. .
1 13.184 A 20g bullet fired into a 4kg wooden block suspended from cords ' M AC and BD penetrates the block at point E, halfway between C " " and D, without hitting cord BD. Determine (a) the maximum
height h to which the block and the embedded bullet will swing
after impact, (I?) the total impulse exerted on the block by the two
cords during the impact. 13.185 A 70g ball B dropped from a height ho = 1.5 m reaches a height
kg = 0.25 m after bouncing twice from identical 210g plates. Plate
A rests directly on hard ground, while plate C rests on a foam
rubber mat. Determine (a) the coefficient of restitution between the ball and the plates, (1)) the height hi of the ball’s first bounce. Fig. ”3.185 842 Kine'i“ °f PGF'iC'W Energy and M°menlum 13.186 Ball B is hanging from an inextensible cord. An identical ballA .
Meth°ds released from rest when it is just touching the cord and drop
through the vertical distance h = 8 in. before striking baler Assuming e = 0.9 and no friction, determine the resulting maxi mum vertical displacement ha of the ball B. ‘  13.187 A 700g sphere A moving with a velocity v0 parallel to the ground
strikes the inclined face of a 2.1kg wedge B which can roll freel
on the ground and is initially at rest. After impact the spherea’lis
observed from the ground to be moving straight up. Knowih.
that the coefficient of restitution between the sphere and the
wedge is e = 0.6, determine (a) the angle 0 that the inclined fac
of the wedge makes with the horizontal, (b) the energy lost du
to the impact. 13.188 When the rope is at an angle of a = 30° the 2lb sphere A has'lga
speed 00 = 2 ft/s. The coefficient of restitution between A and th
4lb wedge B is 0.8 and the length of rope I = 3 ft. The spring
constant has a value of 100 lb/ft and 9 = 20°. Determine the velqé
ity of A and B immediately after the impact. Fig. ”3.187 Fig. pmss
13.189 When the rope is at an angle of a = 30° the 0.5kg sphere A has
a speed 00 = 1.2 m/s. The coefficient of restitution between A arid
the 0.9kg wedge B is 0.7 and the length of rope I = 0.8 m. The spring constant has a value of 500 N/m and 0 = 20°. Determine
the velocity of A and B immediately after the impact. \ Fig. P13.189 1a is
es BS an 13.190 A 2—02 pellet shot vertically from a spring—loaded pistol on the 13.191 REVIEW PROBLEMS surface of the earth rises to a height of 300 ft. The same pellet
shot from the same pistol on the surface of the moon rises to a
height of 1900 ft. Determine the energy dissipated by aerodynamic
drag when the pellet is shot on the surface of the earth. (The
acceleration of gravity on the surface of the moon is 0.165 times
that on the surface of the earth.) An elastic cable is to be designed for bungee jumping from a tower
130 ft high. The specifications call for the cable to be 85 ft long
when unstretched, and to stretch to a total length of 100 ft when
a 600lb weight is attached to it and dropped from the tower.
Determine (a) the required spring constant k of the cable, (19) how
close to the ground a 185lb man will come if he uses this cable
to jump from the tower. 13.192 A 202 hollow steel sphere attached to an 8in. cord can swing about point 0 in a vertical plane. It is subjected to its own weight
and to a force F exerted by a small magnet embedded in
the ground. The magnitude of that force expressed in pounds is
F = 0.1/r2, where r is the distance from the magnet to the sphere
expressed in inches. Knowing that the sphere is released from rest
at A, determine its speed as it passes through point B. Fig. 1913.192 13.193 A satellite describes an elliptic orbit about a planet of mass M. The minimum and maximum values of the distance r from the satellite
to the center of the planet are, respectively, 1‘0 and 13. Use the
principles of conservation of energy and conservation of angular
momentum to derive the relation 1 1_2GM To 1‘1 h2 Where h is the angular momentum per unit mass of the satellite
and G is the constant of gravitation. Fig. mam Fig. P13.193 ...
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