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HW- 8: C h. 19 - Ki neti cs of an ER O- 2D ( Impul se- M omentum)
HW-8: Ch. 19 - Kinetics of an ERO-2D (Impulse-Momentum) Due: 11:59pm on Sunday, July 20, 2014 To understand how points are awarded, read the Grading Policy for Policy for this this as signment.
Principle of Impulse and Momentum Learning Goal: To be able to solve problems involving force, moment, velocity, and time by applying the principle of impulse and momentum to rigid bodies. The principle of impulse and momentum states that the sum of all impulses created by the external forces and moments that act on a rigid body during a time interval is equal to the change in the linear and angular momenta of the body during that time interval. In other words, impulse is the change in momentum. The greater the impulse exerted on a body, the greater the body’s change in momentum. For example, baseball batters swing hard to maximize the impact force and follow through to maximize the impact time. This principle holds true for both linear and angular impulse and momentum. For a rigid-body’s planar motion, the equations for the linear impulse and momentum in the x –y plane plane are given by
Similarly, the equation for the principle of angular impulse and momentum about the z axis, axis, which passes through the rigid-body’s mass center , is giv given en by
Part A - Angular - Angular velocity velo city of the pulley The pulley shown has a moment of inertia = 0.900 , a radius = 0.3 0.300 00
, an and d a mass of 20 20.0 .0
. A cylin cylinde derr is
attached to a cord that is wrapped around the pulley. Neglecting bearing friction and the cord’s mass, express the pulley’s final angular velocity in terms of the magnitude of the cord’s tensi tension, on, (measured in ), 2.00 af after ter the sys system tem is released released from from rest. Use the principle of angular impulse and momentum. Express your answer numerically in radians per second to three significan ignificantt figures.
Hint 1. How to approach the problem 1. Draw a free-bod free-body y diagram of the pulley showing s howing all the forces forces and couple moments that t hat http://session.master ing eng i neer ing .com/myct/assi g nmentPr i ntVi ew?assi g nmentID= 1182016
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HW- 8: C h. 19 - Ki neti cs of an ER O- 2D ( Impul se- M omentum)
produce impulses on the pulley. 2. Expres Express s the final angular velocit elocity, y,
, in terms of
by applying the principle of angular
impulse impuls e and momentum, which st states ates that the final angular momentum, adding the initi initial al angular momentum,
, is obtained by
, and the angular impuls impulses es of moment
during the time interval.
Hint 2. Com Complete plete the free-b ody diagram of the pulley Complete the free-body diagram of the pulley by adding the forces that act on it. Draw the reactions at A ending a t point A and pointing in the positive x and and y directions. directions. Draw the other vectors starting at the dots on the pulley’s circumference. The starting or ending point and orientation of your vectors will be graded. The length of your vectors will not be graded. ANSWER:
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Hint 3. Identify what is needed to apply the principle of angular impulse and momentum Which of the following statements are relevant when applying the principle of angular impulse and momentum momen tum to the pulley? Check all that apply. ANSWER: The initial angular momentum of the pulley is zero. The pulley’s angular momentum is the product of the pulley’s moment of inertia and the angular velocity. The pulley’s angular momentum is the product of the pulley’s mass and the angular velocity. The angular impulse is determined by time integration of the moments about point A during the 2.00 2.0 0 int inter erv val. The final angular momentum of the pulley is zero.
Hint 4. Angular 4. Angular impulse generat ge nerated ed by the th e tension What is the angular impuls impulse e generated by the tensi tension on in terms of the tensi tension’s on’s magnitude,
?
ANSWER:
ANSWER: http://session.master ing eng i neer ing .com/myct/assi g nmentPr i ntVi ew?assi g nmentID= 1182016
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= 0.667
All attempts used; correct answer displayed A change in the angular momentum of a rigid body is caused by an angular impulse acting on the body. The angular impulse is determined by time integration of the moments of all external forces and the applied couple moments.
Part B - Principle of linear impulse and momentum For the same system, determine the final velocity of the cylinder of mass
= 12.0
that is attached to the
pulley. Express your answer to three significant figures and include the appropriate units.
Hint 1. How to approach the problem 1. Draw the free-body diagram of the cylinder showing all the forces that produce impuls es on the cylinder. 2. Express the magnitude of the tension in the cord in terms of the final velocity, , by applying the principle of linear impulse and momentum, which states that the final linear momentum, , is obtained by adding the initial linear momentum, , and the impulses exerted by the tension and the weight during the time interval. 3. Relate the cylinder’s final velocity with the pulley’s final angular velocity using rigid-body kinematics. 4. Solve the simultaneous equations to calculate the final velocity of the cylinder.
Hint 2. Complete the free-body diagram of the cylinder Complete the free-body diagram of the cylinder by adding the forces that act on it. Draw the vectors starting at the black dots. The starting point and orientation of the vectors will be graded. The length of the vectors will not be graded. ANSWER:
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Hint 3. Express the tension in the cord in terms of the final velocity of the cylinder Which of the following is the correct expression for the tension’s magnitude? ANSWER:
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Hint 4. Relate the angular velocity of the pu lley and the velocity of the cylinder Which of the following is the relationship between the angular velocity of the pulley and the velocity of the cylinder? ANSWER:
Hint 5. Identify the final velocity of the cylinder Which is the correct expression for the final velocity of the cylinder? ANSWER:
ANSWER: =
10.7
All attempts used; correct answer displayed A change in the linear momentum of a rigid body is caused by a linear impulse acting on the body. The linear impulse is determined by integrating the external forces with respect to time.
Part C - Principle of angular impulse and momentum applied to the entire system http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1182016
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Determine the mass of block B necessary to cause the 30.0 in 8.00 . The pulley of mass 20.0
block A to change its velocity from 8.00 to 16.0
has a moment of inertia of 0.900
that the pulley rotates about a frictionles s bearing. The coefficient of friction,
and a radius of 0.300
. Assume
, between block A and the surface is
0.250. Apply the principle of angular impulse and momentum to the entire block-pulley system shown. Express your answer to three significant figures and include the appropriate units.
Hint 1. How to approach the problem 1. Draw the impulse and momentum diagrams of the system showing all the forces and couple moments that produce impulses on the pulley and blocks. 2. Apply the principle of angular impulse and momentum to the pulley-block s ystem and determine the block’s mass.
Hint 2. Label the impulse and momentum diagram Label the impulse and momentum diagram. Drag the appropriate labels to their respective targets. ANSWER:
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Hint 3. Identify the equation of angular impulse and momentum Which of the following is the correct equation for angular impulse and momentum? ANSWER:
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ANSWER: = 12.9
All attempts used; correct answer displayed Applying the principle of impulse and momentum to an entire system of connected bodies, rather than to individual bodies, eliminates the need to include the reactive impulses that occur at the connections because they are internal to the system. The equation for the principle of angular impulse and momentum may be written in symbolic form as
Problem 19.16
Part A If the boxer hits the 75
punching bag with an impulse of
, determine the angular velocity of the bag
immediately after it has been hit. Express your answer with the appropriate units. Assume the counterclockwise rotation as positive.
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ANSWER: =
0.328
Correct
Part B Also, find the location
of point
, about which the bag appears to rotate. Treat the bag as a uniform cylinder.
Express your answer with the appropriate units. ANSWER: =
6.25×10−2
Correct
Problem 19.26 The body and bucket of a skid steer loader has a weight of 1970 , and its center of gravity is located at the four wheels has a weight of 105 and a radius of gyration about its center of gravity of 1 .
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. Each of
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Part A If the engine supplies a torque of in
= 105
to each of the rear drive wheels, determine the speed of the loader
= 13 , starting from rest. The wheels roll without slipping.
Express your answer with the appropriate units. ANSWER: =
26.4
All attempts used; correct answer displayed
Problem 19.29 • The car strikes the side of a light pole, which is designed to break away from its base with negligible resistance. From a video taken of the collision it is observed that the pole was given an angular velocity of 62 when was vertical. The pole has a mass of 175 , a center of mass at , and a radius of gyration about an axis perpendicular to the plane of the pole assembly and passing through of = 2.30 .
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Part A Determine the horizontal impulse which the car exerts on the pole at the instant
is essentially vertical.
Express your answer with the appropriate units. ANSWER: = 16.4
Correct
Problem 19.15 The 1.22
tennis racket has a center of gravity at
and a radius of gyration about
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of
= 0.655
.
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Part A Determine the position
where the ball must be hit so that 'no sting' is felt by the hand holding the racket, i.e., the
horizontal force exerted by the racket on the hand is zero. Express your answer with the appropriate units. ANSWER: = 1.43
Correct
± Conservation of Momentum Learning Goal: To be able to describe the motion of rigid bodies by applying the conservation of linear and angular momenta. If the sum of all the linear impulses acting on a system of connected rigid bodies is zero, the linear momentum of the system is conserved. Mathematically, this relationship is expressed as
and is called the conservation of linear momentum. If the sum of all the angular impulses (created by the external forces that act on the system) is negligible or zero, then the angular momentum of a system of connected rigid bodies is conserved about the system's center of mass or about a fixed point. Mathematically, this relationship is expressed as
and is called the conservation of angular momentum.
Part A Which of the following scenarios demonstrate the conservation of either linear or angular momentum? Check all that apply. ANSWER: From opposite sides of a room, two identical balls of putty move toward each other, without friction, at the same velocity and, eventually, they collide; the result is one ball of putty with zero velocity. A penny is dropped from the top of a building and its velocity increases as it falls due to the acceleration from gravity. An ice skater tucks in her arms during a spin and her angular velocity increases. A parent pushes a merry-go-round and, consequently, it spins faster.
Correct
Part B http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1182016
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A pendulum consists of a slender rod, AB, of weight The length of the rod is sphere is 0.500
= 0.250
= 8.00
= 9.40
and a wooden sphere of weight
= 27.4
.
and the radius of the
. A projectile of weight
=
strikes the center of the sphere at a velocity of
= 730
and becomes embedded in the center of
the sphere. What is
, the angular velocity of the
pendulum, immediately after the projectile st rikes the sphere? Express your answer numerically in radians per second to three significant figures.
Hint 1. How to approach the problem Consider the projectile and the pendulum to be part of the same system. Because the projectile exerts an impulse on the pendulum that is equal to but opposite in direction of the impulse that the pendulum exerts on the projectile, these impulses c an be omitted from the analysis. Without any external forces acting on the system, the angular momentum around point A is conserved. Find the angular momentum of the projectile about point A immediately before impact, and then derive an expression for the angular momentum of the system after impact in terms of the sys tem's angular velocity.
Hint 2. Find the initial angular momentum What is
, the initial angular momentum of the system about point A?
Express your answer numerically in slug-squared feet per second to four significant figures.
Hint 1. Find an expression for the initial angular momentum What is
, the initial angular momentum of the system about point A in terms of the following
variables: the weight of the projectile,
; the length of the rod,
; the radius of the sphere,
acceleration due to gravity, ; and the initial velocity of the projectile, Express your answer in terms of
,
,
, , and
; the
?
.
ANSWER: =
ANSWER: = 93.52
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HW-8: Ch. 19 - Kinetics of an ERO-2D (Impulse-Momentum)
Hint 3. Find the total moment of inertia of the system after impact What is
, the total moment of inertia of the system after impact?
Express your answe r in slug-square d fee t to four significant figures.
Hint 1. Find the rod's moment of inertia What is
, the rod's moment of inertia about point A?
Express your answer numerically in slug-squared feet to four significant figures.
Hint 1. Find an expression for the rod's moment of inertia What is
, the rod's moment of inertia about point A, in terms of the following variables: the
rod's length,
; the rod's weight,
; and the acceleration due to gravity, ? Consult your
textbook for the formula of a slender rod's moment of inertia about its end. Express your answer in terms of
,
, and
.
ANSWER: =
ANSWER: = 6.228
Hint 2. Find the sphere' s moment of inertia What is
, the sphere's moment of inertia about point A?
Express your answer numerically in slug-squared feet to four significant figures.
Hint 1. Find an expression for the sphere's moment of inertia What is
, the sphere's moment of inertia about point A, in terms of the following variables:
the rod's length,
; the sphere's radius,
; the sphere's weight,
; and the acceleration
due to gravity, ? Consult your textbook for the moment of inertia of a sphere that rotates about its center. Use the parallel-axis theorem to transfer the moment of inertia of the sphere to the axis about A; this is done by adding the product of the sphere's mass and the square of the distance between the sphere's center and point A to the moment of inertia about an axis through the center of mass. Express your answer in terms of
,
,
, and
.
ANSWER:
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=
ANSWER: = 57.94
Hint 3. Find the projectile's moment of inertia What is
, the projectile's moment of inertia about point A after impact?
Express your answer numerically in slug-squared feet to four significant figures.
Hint 1. Find an expression for the projectile's moment of inertia What is
, the projectile's moment of inertia about point A, in terms of the following variables:
the rod's length,
; the sphere's radius,
; the projectile's weight,
; and the acceleration
due to gravity, ? Express your answer in terms of
,
,
, and
.
ANSWER: =
ANSWER: =
1.057
ANSWER: = 65.22
Hint 4. Find an expression for the a ngular velocity What is an expression for moment of inertia,
, the angular velocity of the system after impact, in terms of the system's total
, and the initial angular momentum,
Express your answer in terms of
and
, of the system.
.
ANSWER:
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=
ANSWER: = 1.43
All attempts used; correct answer displayed
Part C What is
, the maximum angle measured from the vertical that the pendulum will swing, after the projectile impacts
the pendulum? Express your answer numerically in degrees to three significant figures.
Hint 1. How to approach the problem After impact, the energy of the pendulum and lodged projectile is conserved. To solve for the maximum angle the pendulum will swing, use conservation of energy: where
is the initial kinetic energy,
is the initial potential energy,
is the final kinetic energy, and
is the final potential energy of the sys tem. In this situation, the initial conditions are immediately after impact, and the final conditions are at the maximum angle. Set the initial potential energy of the sy stem to zero and express the final potential energy term as a function of the maximum angle , and solve for .
Hint 2. Find What is
, the kinetic energy of the system immediately after impact
, the kinetic energy of the system immediately after the projectile's impact?
Express your answe r nume rically in foot-pounds to four significant figures.
Hint 1. Finding the kinetic energy of the system after impact Immediately after impact, the kinetic energy of the system is due to the rotational motion about point A:
where
is the moment of inertia of the system, found in Part B to be 65.22
angular velocity of the system, found in Part B to be 1.43
, and
is the
.
ANSWER: =
67.04
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Hint 3. Find an expression for the p rojectile's final potential energy Find an expression for the length of the rod;
, the projectile's final potential energy, in terms of
, the projectile's weight;
,
, the radius of the sphere; and , the maximum angle the pendulum sweeps after
impact. Express your answer in terms of
,
,
, and
.
Hint 1. Finding the change in height of the projectile The projectile originates at a distance the projectile is
below point A. At the peak of the pendulum motion,
below point A. The change in height of the projectile is the
difference in these quantities.
ANSWER: =
Hint 4. Find an expression for the sphere' s final potential energy Find an expression for length of the rod;
, the sphere's final potential energy, in terms of
, the sphere's weight;
, the
, the radius of the sphere; and , the maximum angle the pendulum sweeps after impact.
Express your answer in terms of
,
,
, and
.
Hint 1. Finding the change in height of the sphere The sphere's center originates at a distance
below point A. At the peak of the pendulum's
motion, the sphere's center is
below point A. The change in height of the sphere is
the difference in these quantities.
ANSWER: =
Hint 5. Find an expression for the rod' s final potential energy Find an expression for weight;
, the rod's final potential energy, in terms of the following variables:
, the rod's
, the length of the rod; and , the maximum angle that the pendulum sweeps after impact.
Express your answer in terms of
,
, and
.
Hint 1. Finding the change in height of the rod The rod's center of mass originates at a distance motion, the rod's center is
below point A. At the peak of the pendulum
below point A. The change in height of the rod is the
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difference in these quantities.
ANSWER:
=
ANSWER: = 41.4
Correct
Problem 19.41 • Two children
and , each having a mass of 30 , sit at the edge of the merry-go-round which rotates at . Excluding the children, the merry-go-round has a mass of 180 and a radius of gyration
.
Part A Determine the angular velocity of the merry-go-round if 2
jumps off horizontally in the
direction with a speed of
, measured relative to the merry-go-round. Neglect friction and the size of each child.
Express your answer with the appropriate units. ANSWER: =
2.43
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Correct
Part B What is the merry-go-round's angular velocity if
then jumps off horizontally in the
direction with a speed of 2
, measured relative to the merry-go-round? Express your answer with the appropriate units. ANSWER: =
2.96
Correct
Problem 19.50 The rigid 30-
plank is struck by the 15-
hammer head
.
Part A Just before the impact the hammer is gripped loosely and has a vertical velocity of 75 restitution between the hammer head and the plank is block
=
. If the coefficient of
, determine the maximum height attained by the 50-
. The block can slide freely along the two vertical guide rods. The plank is initially in a horizontal
position. Express your answer with the appropriate units. ANSWER: = 4.99
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Correct
Problem 19.54 The 4- rod hangs in the vertical position. A 2, strikes the rod at its end .
block, sliding on a smooth horizontal surface with a velocity of 12
Part A Determine the direction of the velocity of the block immediately after the collision. The coefficient of restitution between the block and the rod at is = . ANSWER: to the left to the right
Correct
Part B Determine the magnitude of the velocity of the block immediately after the collision. Express your answer with the appropriate units. ANSWER: =
3.36
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Problem 19.47 The target is a thin 5 circular disk that can rotate freely about the the target at and becomes embedded in it.
axis. A 23 bullet, traveling at 615
, strikes
Part A Determine the angular velocity of the target after the impact. Initially, it is at rest. Express your answer with the appropriate units. Assume the counterclockwise rotation as positive. ANSWER: =
24.9
Correct
Problem 19.55 The pendulum consists of a 10-
sphere and 4-
rod.
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Part A If it is released from rest when
= 90 , determine the angle
of rebound after the sphere strikes the floor. Take
= 0.76. Express your answer with the appropriate units. ANSWER: = 35.3
Correct Score Summary: Your score on this assignment is 96.7%. You received 8 out of a possible total of 9 points, plus 0.71 points of extra credit.
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