Download 1 References Slides also Available at Some Tricks Dynamics

References
Website: https://moodle.umn.edu/course/view.php?id=7988
Sections 14-17
FE Supplied-Reference Handbook, Page 54-61
FE Review – Dynamics 1/4
Kinematics
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By Dr. Debao Zhou
Kinematics
Kinetics
Kinetics of rotational motion
Energy and work
Department of Mechanical & Industrial Engineering
University of Minnesota Duluth
1
Slides also Available at
•
Some Tricks
Kinematics
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http://www.d.umn.edu/~dzhou/01Kenematics1.pdf
Kinetics
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http://www.d.umn.edu/~dzhou/02Kinetics1.pdf
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Kinetics of rotational motion
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Energy and work
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http://www.d.umn.edu/~dzhou/03KineticsofRotational1.pdf
Only the results
Unit: make the unit consistent
Confused data
Keep your drafts
• Concepts/Formula/Multi-ways, with the aid
of index
• Try to draw a diagram, illustration, etc.
http://www.d.umn.edu/~dzhou/04Energy1.pdf
– Free body diagram
3
Dynamics - Scope
4
Kinematics: Position, Velocity and Acceleration
•
Dynamics - Study of moving objects
•
Kinematics: Motion (Position, velocity and acceleration), Independent of force
•
Kinetics:
•
Energy and Work
•
Translational motion
– Particle kinematics
– Rigid body kinematics
International System of Units (SI):
http://en.wikipedia.org/wiki/International_System_of_Units
US Units (U.S.)
http://en.wikipedia.org/wiki/United_States_customary_units
Relationship: http://www.about.ch/various/unit_conversion.html
– Force and mass for translational motion (particle, body)
– Torque and moment of inertia for rotational motion (body)
– Capability of the mass to do work: All kinds of energy
– Position, velocity and acceleration, mass, force, friction
– Momentum, impact, work, energy
– And change with time
•
Rotational motion
– Angle, angular velocity and angular acceleration, inertia, moment (torque), friction
– Angular momentum, impulse, impact, work, energy
– And change with time
•
Combination of translational motion and rotational motion
– Multi-body: Instant velocity center. etc
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6
1
Kinematics: Position, Velocity and Acceleration
Position, Velocity and Acceleration
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Constant Acceleration
Linear motion
Constant
8
Circular motion
• They have the same relationship as linear system
Rotational motion
Constant
• Relation between linear and rotation variables
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Project Motion
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Example -1
• Problems 1-2 refer to a particle whose curvilinear motion is
represented by the equation s = 20t + 4t2 – 3t3.
• Trajectory
– Vertical
acceleration
Only
• Acceleration
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2
Example -2
Example -2
• A motorist is traveling at 70 km/ h when he sees a traffic light in an
intersection 250 m ahead turn red. The light 's red cycle is 15 s. The
motorist wants to enter the intersection without stopping his vehicle,
just as the light turns green.
What uniform deceleration of the vehicle will just put the motorist in
the intersection when the light turns green?
(A) 0.18 m/s2
250 m @ Time 15 s
(B) 0.25 m/s2
(C) 0.37 m/s2
(D) 1.3 m/s2
V = 70 km/ h
0
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Example -3
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Example -4
• The position (in radians) of a car traveling around a curve
is described by the following function of time (in seconds).
(t) = t3 - 2t2 - 4t + 10
What is the angular velocity at t = 3 s?
(A) -16 rad/ s
(B) - 4 rad / s
(C) 11 rad/ s
(D) 15 rad/ s
• A flywheel rotates at 7200 rev/min when the power is
suddenly cut off. The flywheel decelerates at a constant
rate of 2.1 rad/s2 and comes to rest 6 min later. How
many revolutions does the flywheel make before coming
to rest?
(A) 18 000 rev
What are given?
(B) 22 000 rev
Formula?
Unit?
(C) 72 000 rev
Solution
(D) 390 000 rev
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Example -4
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Example 5
• Rigid link AB is 12 m long. It rotates counterclockwise about point A
at 12 rev/min. A thin disk with radius 1. 75 m is pinned at its center to
the link at point B. The disk rotates counterclockwise at 60 rev/ min
with respect to point B. What is the maximum tangential velocity seen
by any point on the disk?
(A) 6 m/s
(B) 26 m/s
(C) 33 m/ s
(D) 45 m/s
What are given?
Formula?
Unit?
Solution
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3
Example 5
Example -6
• A projectile is fired from a cannon with an initial velocity of
1000 m/ s and at an angle of 30° from the horizontal.
What distance from the cannon will the projectile strike
the ground if the point of impact is 1500 m below the
point of release?
(A) 8200 m
(B) 67300 m
(C) 78200 m
(D) 90800 m
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What are given?
Formula?
Unit?
Solution
FE Review – Dynamics 2/4
Kinetics
By Dr. Debao Zhou
Department of Mechanical & Industrial Engineering
University of Minnesota Duluth
21
Kinetics: Scope
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Kinetics: SI and US Units
• Kinetics: motion and force that cause motion
• Momentum
– Linear momentum and angular momentum
– Law of conservation of momentum
• Newton’s first and second law of motion
– Acceleration is zero or not
• Weight
– The force the object exerts due to its position in a gravitational field
– gc is the gravitational constant, approximately 32.2 lbm ft / lbf-sec2
.
• Friction
• Particles: tangential and normal components
• Free Vibration
4
Linear Momentum or Momentum
Kinetics: Unit
SI: N-sec
• Definition:
?
U.S.: lbf-sec
• Law of conservation of momentum
– The linear momentum is unchanged if no
unbalanced forces act on the particle.
– This does not prohibit the mass and velocity
from changing. However, only the product of
mass and velocity is constant.
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Newton's Law of Motion
Kinetics of Particle
• Newton's first law of motion
– A particle will remain in a state of rest or will continue to
move with constant velocity unless an unbalanced
external force acts on it.
• Rectangular Coordinates
• Tangential and Normal Components
Constant Fx
• Constant velocity unless an unbalanced external force acts on it.
• Law of conservation of momentum
• Newton's second law of motion
– The acceleration of a particle is directly proportional to
the force acting on it and is inversely proportional to the
particle mass.
• Radial and Transverse Components
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Free vibration
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Friction
• Natural (or free) vibration.
• Forced vibration.
•
•
•
•
• Natural frequency
– Or angular frequency
Always resists motion
Parallel to the contacting surfaces
Dynamic friction
Static friction
Internal force
75%
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Example -1
Example -2
• A car is pulling a trailer at 100 km/ h. A 5 kg cat riding on the roof of
the car jumps from the car to the trailer. What is the change in the
cat's momentum (when it is on the car and trailer)?
(A) -25 N·s (loss)
(B) 0 N·s
System? What is the whole thing
you are considering?
(C) 25 N-s (gain)
(D) 1300 N- s (gain)
• A car with a mass of 1530 kg tows a trailer (mass of 200 kg) at 100
km/ h. What is the total momentum of the car-trailer combination?
(A) 4 600 N·s
(B) 22 000 N·s
(C) 37 000 N·s
(D) 48 000 N·s
• The law of conservation of momentum states that the linear
momentum is unchanged if no unbalanced forces act on an object.
This does not prohibit the mass and velocity from changing; only the
product of mass and velocity is constant. In this case, both the total
mass and the velocity are constant . Thus, there is no change.
What are given?
Formula?
Unit?
Solution
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Example -3
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Example -4
• For which of the following situations is the net
force acting on a particle necessarily equal to
zero?
A. The particle is travelling at constant velocity
around a circle Vector!!!
B. The particle has constant linear momentum
C. The particle has constant kinetic energy
D. The particle has constant angular momentum
• A 3500 kg car accelerates from rest. The constant forward tractive
force of the car is 1000 N, and the constant drag force is 150 N. What
distance will the car travel in 3s?
(A) 0.19 m
(B) 1.1 m
(C) 1.3 m
(D) 15 m
1000N
150N
3500 kg
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Example - 5
Example -6
• A 5 kg block begins from rest and slides down an inclined plane. After
4s, the block has a velocity of 6 m/s.
If the angle of inclination is 45°; how far has the block traveled after 4s?
(A) 1.5 m
(B) 3 m
(C) 6 m
(D) 12 m
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• What is the coefficient of friction between the plane and the block?
(A) 0.15
(B) 0.22
(C) 0.78
(D) 0.85
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6
Example -7
Example -8
• A spring has a constant of 50 N/ m. The spring is hung vertically, and
a mass is attached to its end. The spring end displaces 30 cm from
its equilibrium position. The same mass is removed from the first
spring and attached to the end of a second (different) spring, and the
displacement is 25 cm. What is the spring constant of the second
spring?
(A) 46 N/m
(B) 56 N/m
(C) 60 N/m
(D) 63 N/m
• A constant force of 750 N is applied through a pulley system to lift a
mass of 50 kg as shown. Neglecting the mass and friction of the
pulley system, what is the acceleration of the 50 kg mass?
(A) 5.20 m/s2; (B) 8.72 m/ s2; (C) 16.2 m/s2; (D) 20.2 m/s2
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Example -9
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Example -10
What is the period of a pendulum that passes the center point 20 times a
minute?
(A) 0.2 s
(B) 0.3 s
(C) 3 s
(D) 6 s
• A mass of 10 kg is suspended from a vertical spring with a spring
constant of 10 N/ m. What is the period of vibration?
(A) 0.30 s
(B) 0.60 s
(C) 0.90 s
(D) 6.3 s
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Kinetics of Rotational Motion - Scope
• Mass moment of inertia
– Parallel axis theorem
– Radius of gyration
• Planar motion of rigid body
FE Review – Dynamics 3/4
– Angular momentum and moment (torque)
– Instant center (velocity)
Kinetics of Rotational Motion
• Centrifugal force
• Banking angle
• Torsion vibration
By Dr. Debao Zhou
Department of Mechanical & Industrial Engineering
University of Minnesota Duluth
41
7
Mass Moment of Inertia
Remember the definition and formula – 3D ( 2D)
http://en.wikipedia.org/wiki/Moment_of_inertia
• Respect to the x-, y- , and z-axes
• Centroid mass moment of inertia
– When the origin of the axes coincides with
the object 's center of gravity


  x


 y dm
I x   y 2  z 2 dm
I y   x 2  z 2 dm
Iz
2
2
• Parallel axis theorem
I any parallel axis  I c  md 2
• Radius of gyration
The distance from the rotational axis at
which the object’s entire mass could be
located without changing the mass moment of inertia.
I  r 2m
r I
m
• Table 16.1 (at the end of this chapter) lists the mass moments of
inertia and radii of gyration for some standard shapes.
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Rotation About a Fixed Axis
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Instantaneous Center of Rotation
• Rotation describes a motion in which all particles within the body
move in concentric circles about the axis of rotation.
• Angular momentum taken about a point O (fixed point)
– Moment of the linear momentum vector
• For angular velocities, the body seems to rotate
about a fixed instantaneous center
• Lines drawn perpendicular to these two velocities
will intersect at the instantaneous center
• Moment (torque), M
Constant M
?
• Conservation
•
•
Similar to Newton’s first/second law
From translational motion to rotational motion
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Centrifugal Force
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Banking of Curves
• The force associated with the normal acceleration is
known as the centripetal force
– A real force
• The so-called centrifugal force is an apparent force on the
body directed away from the center of rotation
– Acceleration force
• http://farside.ph.utexas.edu/t
eaching/301/lectures/node92.
html
• For small banking angles, the
maximum frictional force is
F f   s N   sW
• If the roadway is banked so
that friction is not required to
resist the centrifugal force,
the superelevation angle, ,
can be calculated from
tan( ) 
47
vt2
gr
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8
Torsional Free Vibration
Example - 1
• A 50 kg cylinder has a height of 3m and a radius of 50cm. The
cylinder sits on the x-axis and is oriented with its major axis parallel
to the y-axis. What is the mass moment of inertia about the x-axis?
(A) 4.1 kg·m2
(B) 16 kg·m2
(C) 41 kg·m2
(D) 150 kg·m2
• Dynamic equation
  n2  0
• Solution
 0 
 sin n t
 n 
 (t )   0 cos n t  
• Natural frequency
n2 
kt GJ

I
LI
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Example - 2
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Example -3
• A 3 kg disk with a diameter of 0.6 m is rigidly attached at point B to a
1 kg rod 1 m in length. The rod-disk combination rotates around point
A. What is the mass moment of inertia about point A for the
combination?
(A) 0.47 kg·m2
(B) 0.56 kg ·m2
(C) 0.87 kg·m2
(D) 3.7 kg·m2
Why does a spinning ice skater 's angular velocity increase as
she brings her arms in toward the body:
(A) Her mass moment of inertia is reduced.
(B) Her angular momentum is constant.
(C) Her radius of gyration is reduced.
(D) all of the above
As the skater brings her arms in, her radius of gyration and mass
moment of inert is decrease. However, in the absence of
friction, her angular momentum h, is constant . From

h
I
Since angular velocity, , is inversely proportional to the mass
moment of inertia, the angular velocity increases when the
mass moment of inertia decreases.
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Example -4
Example -4
• A wheel with a radius of 0.75 m starts from rest and accelerates
clockwise. The angular acceleration (in rad/s2) of the wheel is defined
by  = 6t - 4. What is the resultant linear acceleration of a point on
the wheel rim at t = 2 s?
(A) 6 m/ s2
(B) 12 m/ s2
(C) 13 m/ s2
(D) 18 m/ s2
B
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Example -5
Example -5
• A uniform rod (AB) of length L and weight W is pinned at point C and
restrained by cable OA. The cable is suddenly cut. The rod starts to
rotate about point C, with point A moving down and point B moving
up. What is the instantaneous linear acceleration of point B?
Free body diagram
55
Example 6
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Example 6
• A wheel with a 0.75 m radius has a mass of 200 kg. The wheel is
pinned at its center and has a radius of gyration of 0.25 m. A rope is
wrapped around the wheel and supports a hanging 100 kg block.
When the wheel is released, the rope begins to unwind. What is the
angular acceleration of the wheel?
(A) 5.9 rad/s2
(B) 6.5 rad/s2
(C) 11 rad/s2
(D) 14 rad/s2
Free body diagram
F
F
mblockg
57
Example -7
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Example - 8
• Two 2 kg blocks are linked as shown. Assuming
that the surfaces are frictionless. what is the
velocity of block B if block A is moving at a speed
of 3 m/ s?
A) 0 m/s; (B) 1.30 m/s; (C) 1.73 m/s; (D) 5.20 m/s
• A disk rolls along a flat surface at a constant speed of 10 m/s. Its
diameter is 0.5 m. At a particular instant, point P on the edge of the
disk is 45° from the horizontal. What is the velocity of point P at that
instant?
(A) 10.0 m/s, (B) 15.0 m/s; (C) 16 .2 m/s; (D) 18.5 m/s
• The instantaneous center of rotation for the slider
rod assembly can be found by extending
perpendiculars from the velocity vectors, as
shown. Both blocks can be assumed to rotate
about point C with angular velocity .
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Example 9
Example 10
• An automobile travels on a perfectly horizontal, unbanked circular
track of radius r. The coefficient of friction between the tires and the
track is 0.3. If the car's velocity is 10 m/s, what is the smallest radius
it may travel without skidding?
• (A) 10 m; (B) 34 m; (C) 50 m; (D) 68 m
• Traffic travels at 100 km/ h around a banked highway curve with a
radius of 1000 m. What banking angle is necessary such that friction
will not be required to resist the centrifugal force?
(A) 1.4°; (B) 2.8°; (C) 4.5°; (D) 46°
• Since there is no friction force , the superelevation angle, , can be
determined directly.
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Example 11
A torsional pendulum consists of a 5 kg uniform disk with a diameter of
50 cm attached at its center to a rod 1.5 m in length. The torsional spring
constant is 0.625 N·m/ rad. Disregarding the mass of the rod, what is the
natural frequency of the torsional pendulum?
(A) 1.0 rad/ s; (B) 1.2 rad/ s; (C) 1.4 rad/ s; (D) 2.0 rad/ s
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Energy and Work - Scope
• Definition of energy, work, and power
– Kinetic Energy
– Potential Energy
– Elastic Potential Energy
FE Review – Dynamics 4/4
• Energy conservation principle
• Linear impulse
• Impact
Energy and Work
By Dr. Debao Zhou
Department of Mechanical & Industrial Engineering
University of Minnesota Duluth
65
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Energy and Work
Energy
• Kinetic energy is the sum of the translational and
T  mv 2 / 2
rotational forms
• The energy of a mass represents the
capacity of the mass to do work
– Linear kinetic energy
– Rotational kinetic energy
– Mechanical
– Positive, scalar quantity
– Definition:
– Signed, scalar quantity
– Positive or negative
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• Elastic potential energy
1 2
kx
2
U 2  U1  k ( x22  x12 )
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Linear Impulse
• A vector quantity equal to the change in (linear)
momentum.
• Work-energy principle
T2  U 2  T1  U1  W12
?
• Energy Conservation Principle: Law of conservation of
energy
• Impulse-momentum principle
– Ti and Ui are, respectively, the kinetic and potential energy of a
particle at state i, if no external work has been done, then
 E  constant
RKE 
PE 
Energy Conservation Principle
W  E2  E1
1 2
I
2
– A form of mechanical energy possessed by a mass due
to its relative position in a gravitational field.
PE  mgh
[SI]
• Potential energy
• Thermal, electrical and magnetic, etc.
• Work, W, is the act of changing the energy
of a mass W   F  dr
T2  T1  m(v22  v12 ) / 2
Constant F
– The change in momentum is equal to the impulse
– Newton’s second law
T2  U 2  T1  U 1
– Energy cannot be created or destroyed
– Energy can be transformed into different forms
– The total energy of the mass is equal to the sum of the potential
(gravitational and elastic) and kinetic energies
• Conservation of linear impulse when F=0
• Newton’s first law
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Impact
• In an impact or collision, contact is very brief, and the effect of external
forces is insignificant. Therefore, momentum is conserved. Even though
energy may be lost through heat generation and deforming the bodies.
– An inelastic impact if kinetic energy is lost .
– The impact is said to be perfectly inelastic or perfectly plastic if the two particles stick
together and move on with the same final velocity.
– The impact is said to be an elastic impact only if kinetic energy is conserved.
Example - 0
• A 1500 kg car traveling at 100 km/h is towing a 250 kg trailer. The
coefficient of friction between the tires and the road is 0.8 for both the
car and trailer.
What energy is dissipated by the brakes if the car and trailer are braked
to a complete stop?
(A) 96 kJ; (B) 385 kJ; (C) 579 kJ; (D) 675 kJ;
• Coefficient of restitution, e:
– The ratio of relative velocity differences along a mutual straight line.
– The collision is inelastic if e < 1.0, perfectly inelastic if e = 0, and
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elastic if e = 1.0
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12
Example -1
Example -1
Problems 1-4 refer to the following situation.
• The mass m in the following illustration is guided by
the frictionless rail and has a mass of 40kg.
• The spring constant, k, is 3000 N/m.
• The spring is compressed sufficiently and released,
such that the mass barely reaches point B.
2. What is the kinetic energy of the mass at point A?
(A) 19.8 J
(B) 219 J
(C) 392 J
(D) 2350 J
1. What is the initial spring compression?
(A) 0.96 m
(B) 1.3 m
(C) 1.4 m
(D) 1.8 m
73
Example -1
74
Example -1
3. What is t he velocity of the mass at point A?
(A) 3.13 m/s
(B) 4.43 m/ s
(C) 9.80 m/s
(D) 19.6 m/ s
4. What is the energy stored in the spring if the spring is
compressed 0.5 m?
(A) 375 J
(B) 750 J
(C) 1500 J
(D) 2100 J
75
Example -2
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Example -3
A 12 kg aluminum box is dropped from rest onto a large wooden beam.
The box travels 20 cm before contacting the beam. After impact, the box
bounces 5 cm above the beam's surface. What impulse does the beam
impart on the box?
A 3500 kg car traveling at 65 km/ h skids and hits a wall 3s later. The
coefficient of friction between the tires and the road is 0.60.
Assuming that the speed of the car when it hits the wall is 0.20 m/s, what
energy must the bumper absorb in order to prevent damage to the car?
(A) 70 J
(B) 140 J
(C) 220 J
(D) 360 kJ
(A) 8.6 N·s
(B) 12 N·s
(C) 36 N·s
(D) 42 N·s
77
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13
Example -4
Example -5
A 60000 kg railcar moving at 1 km/h is instantaneously coupled to a
stationary 40000 kg railcar. What is the speed of the coupled cars?
(A) 0.40 km/ h
(B) 0.60 km/ h
(C) 0.88 km/ h
(D) 1.0 km/ h
A 3500 kg car traveling at 65 km/ h skids and hits a wall 3 s later. The
coefficient of friction between the tires and the road is 0.60.
What is the speed of the car when it hits the wall?
(A) 0.14 m/s; (B) 0.40 m/s; (C) 5.1 m/ s; (D) 6.2 m/ s
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Example -6
A hockey puck traveling at 30 km/h hits a massive wall at an angle of
30° from the wall. What are its final velocity and deflection angle if the
coefficient of restitution is 0.63?
(A) 9.5 km/ h at 30°
(B) 19 km/h at 30°
(C) 28 km/h at 20°
(D) 30 km/ h at 20°
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