hw-8_ ch

11/7/2014 HW-8: Ch. 19 - Kinetics of an ERO-2D (Impulse-Momentum)

http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1182016 1/20

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 this assignment.

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 andmomentum to rigid bodies.

The principle of impulse and momentum states that the sum of all impulses created by the external forces and momentsthat act on a rigid body during a time interval is equal to the change in the linear and angular momenta of the bodyduring that time interval. In other words, impulse is the change in momentum.The greater the impulse exerted on a body, the greater the bodys change in momentum. For example, baseball battersswing 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-bodys planar motion, the equations for the linear impulse and momentum in the xy plane are given by

Similarly, the equation for the principle of angular impulse and momentum about the z axis, which passes through therigid-bodys mass center , is given by

Part A - Angular velocity of the pulley

The pulley shown has a moment of inertia = 0.900 , a radius

= 0.300 , and a mass of 20.0 . A cylinder is

attached to a cord that is wrapped around the pulley.Neglecting bearing friction and the cords mass, expressthe pulleys final angular velocity in terms of themagnitude of the cords tension, (measured in ),

2.00 after the system is released from rest. Use the

principle of angular impulse and momentum.

Express your answer numerically in radians persecond to three significant figures.

Hint 1. How to approach the problem

1. Draw a free-body diagram of the pulley showing all the forces and couple moments that

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produce impulses on the pulley.

2. Express the final angular velocity, , in terms of by applying the principle of angular

impulse and momentum, which states that the final angular momentum, , is obtained by

adding the initial angular momentum, , and the angular impulses of moment

during the time interval.

Hint 2. Complete the free-body 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 at point A and pointing in the positive x and y directions. Draw theother vectors starting at the dots on the pulleys circumference. The starting or ending point andorientation of your vectors will be graded. The length of your vectors will not be graded.

ANSWER:

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 andmomentum to the pulley?

Check all that apply.

ANSWER:

Hint 4. Angular impulse generated by the tension

What is the angular impulse generated by the tension in terms of the tensions magnitude, ?

ANSWER:

ANSWER:

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The initial angular momentum of the pulley is zero.

The pulleys angular momentum is the product of the pulleys moment of inertia and the angularvelocity.

The pulleys angular momentum is the product of the pulleys mass and the angular velocity.

The angular impulse is determined by time integration of the moments about point A during the2.00 interval.

The final angular momentum of the pulley is zero.

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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. Theangular impulse is determined by time integration of the moments of all external forces and the applied couplemoments.

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.


1. Draw the free-body diagram of the cylinder showing all the forces that produce impulses on thecylinder.

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 linearmomentum, , 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 cylinders final velocity with the pulleys 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 begraded. The length of the vectors will not be graded.

ANSWER:

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 tensions magnitude?

ANSWER:

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Hint 4. Relate the angular velocity of the pulley and the velocity of the cylinder

Which of the following is the relationship between the angular velocity of the pulley and the velocity of thecylinder?

ANSWER:

Hint 5. Identify the final velocity of the cylinder

Which is the correct expression for the final velocity of the cylinder?

ANSWER:

ANSWER:


A change in the linear momentum of a rigid body is caused by a linear impulse acting on the body. The linearimpulse is determined by integrating the external forces with respect to time.

Part C - Principle of angular impulse and momentum applied to the entire system

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Determine the mass of block B necessary to cause the 30.0 block A to change its velocity from 8.00 to 16.0

in 8.00 . The pulley of mass 20.0 has a moment of inertia of 0.900 and a radius of 0.300 . Assume

that the pulley rotates about a frictionless bearing. The coefficient of friction, , 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 andinclude the appropriate units.


1. Draw the impulse and momentum diagrams of the system showing all the forces and couplemoments that produce impulses on the pulley and blocks.

2. Apply the principle of angular impulse and momentum to the pulley-block system anddetermine the blocks mass.

Hint 2. Label the impulse and momentum diagram

Label the impulse and momentum diagram.

Drag the appropriate labels to their respective targets.

ANSWER:

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:


Applying the principle of impulse and momentum to an entire system of connected bodies, rather than toindividual bodies, eliminates the need to include the reactive impulses that occur at the connections becausethey 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:

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:

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 . Each ofthe four wheels has a weight of 105 and a radius of gyration about its center of gravity of 1 .

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Part A

If the engine supplies a torque of = 105 to each of the rear drive wheels, determine the speed of the loader

in = 13 , starting from rest. The wheels roll without slipping.


ANSWER:


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 avideo 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.


ANSWER:

Correct

Problem 19.15

The 1.22 tennis racket has a center of gravity at and a radius of gyration about of = 0.655 .

= 16.4 L/T

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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.


ANSWER:

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 thesystem 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 forcesthat act on the system) is negligible or zero, then the angular momentum of a system of connected rigid bodies isconserved 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:

Correct

Part B

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From opposite sides of a room, two identical balls of putty move toward each other, without friction, at thesame 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 accelerationfrom 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.

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A pendulum consists of a slender rod, AB, of weight = 9.40 and a wooden sphere of weight = 27.4 .

The length of the rod is = 8.00 and the radius of the

sphere is = 0.250 . A projectile of weight =

0.500 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 strikes thesphere?

Express your answer numerically in radians persecond to three significant figures.


Consider the projectile and the pendulum to be part of the same system. Because the projectile exerts animpulse on the pendulum that is equal to but opposite in direction of the impulse that the pendulum exertson the projectile, these impulses can be omitted from the analysis. Without any external forces acting onthe system, the angular momentum around point A is conserved. Find the angular momentum of theprojectile about point A immediately before impact, and then derive an expression for the angular momentumof the system after impact in terms of the system'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, ; the

acceleration due to gravity, ; and the initial velocity of the projectile, ?

Express your answer in terms of , , , , and .

ANSWER:

ANSWER:

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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 answer in slug-squared feet 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:

Hint 2. Find the sphere's moment of inertia

What is , the sphere's moment of inertia about point A?


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 sphereto the axis about A; this is done by adding the product of the sphere's mass and the square ofthe distance between the sphere's center and point A to the moment of inertia about an axisthrough the center of mass.

Express your answer in terms of , , , and .

ANSWER:

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ANSWER:

Hint 3. Find the projectile's moment of inertia

What is , the projectile's moment of inertia about point A after impact?


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, ?


ANSWER:

ANSWER:

ANSWER:

Hint 4. Find an expression for the angular velocity

What is an expression for , the angular velocity of the system after impact, in terms of the system's total

moment of inertia, , and the initial angular momentum, , of the system.

Express your answer in terms of and .

ANSWER:

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ANSWER:


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.


After impact, the energy of the pendulum and lodged projectile is conserved. To solve for the maximum anglethe 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 system. In this situation, the initial conditions are immediately afterimpact, and the final conditions are at the maximum angle. Set the initial potential energy of the system tozero and express the final potential energy term as a function of the maximum angle , and solve for .

Hint 2. Find , the kinetic energy of the system immediately after impact

What is , the kinetic energy of the system immediately after the projectile's impact?

Express your answer numerically 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 pointA:

where is the moment of inertia of the system, found in Part B to be 65.22 , and is the

angular velocity of the system, found in Part B to be 1.43 .

ANSWER:

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Hint 3. Find an expression for the projectile's final potential energy

Find an expression for , the projectile's final potential energy, in terms of , the projectile's weight; ,

the length of the rod; , the radius of the sphere; and , the maximum angle the pendulum sweeps after

impact.


Hint 1. Finding the change in height of the projectile

The projectile originates at a distance below point A. At the peak of the pendulum motion,

the projectile is 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 , the sphere's final potential energy, in terms of , the sphere's weight; , the

length of the rod; , the radius of the sphere; and , the maximum angle the pendulum sweeps after impact.


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 , the rod's final potential energy, in terms of the following variables: , the rod's

weight; , 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 below point A. At the peak of the pendulum

motion, the rod's center is below point A. The change in height of the rod is the

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difference in these quantities.

ANSWER:

ANSWER:

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 jumps off horizontally in the direction with a speed of

2 , measured relative to the merry-go-round. Neglect friction and the size of each child.


ANSWER:

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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?


ANSWER:

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 . If the coefficient of

restitution between the hammer head and the plank is = , determine the maximum height attained by the 50-

block . The block can slide freely along the two vertical guide rods. The plank is initially in a horizontal

position.


ANSWER:

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Correct

Problem 19.54

The 4- rod hangs in the vertical position. A 2- block, sliding on a smooth horizontal surface with a velocity of 12, strikes the rod at its end .

Part A

Determine the direction of the velocity of the block immediately after the collision. The coefficient of restitutionbetween the block and the rod at is = .

ANSWER:

Correct

Part B

Determine the magnitude of the velocity of the block immediately after the collision.


ANSWER:

Correct

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Problem 19.47

The target is a thin 5 circular disk that can rotate freely about the axis. A 23 bullet, traveling at 615 , strikes

the target at and becomes embedded in it.

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:

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.


ANSWER:

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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= 35.3 J

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