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8/11/2019 Rotational Motion of Solid Objects 8.1-8.3
http://slidepdf.com/reader/full/rotational-motion-of-solid-objects-81-83 1/46
Announcements Review Quantities in Rotational Motion Center of Mass Torque Final Questions
Chapter 8
Rotational Motion of Solid Objects
Rotational Motion of Solid Objects
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Reading Assignment
Read sections 8.4 - 8.5
Homework Assignment 5
Homework for Chapters 6 and 8 (due Tuesday, October 5)Chapter 6: Q6, Q13, Q24, Q32, E10, E14Chapter 8: Q6, Q12, Q26, E8, E12, E18
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Physics Concept: Energy
Energy (scalar): The ability to do work. Energy is a physical quantity that can be measured, though its value
depends upon the inertial frame of reference (SI units: joules; 1 J = 1 N · m = 1 kg · m2/s2).
The Conservation of Energy
Energy can be neither created nor destroyed, it can only be changed from one form to another or transferred fromone body to another. The total amount of energy is always the same.
Types of energy
Kinetic energy: the energy an object possesses due to its motion
K =1
2mv
2
Potential energy: the energy stored in the forces between or within objects.
Gravitational potential energy: the energy stored in the gravitational forces between an object andthe Earth
U g = mgh
Elastic potential energy: the energy in the forces within a distorted elastic object
U e =1
2kx
2
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Scenario
During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one anotherwith large catapults that are made with surgical hose mounted on a window frame. A 0.5 kg balloon filled withdyed water is placed in a pouch attached to the hose, which is then stretched through the width of the room.Assume that the stretching of the hose obeys Hooke’s law and has a spring constant of 100 N/m.
Rotational Motion of Solid Objects
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Scenario
During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one anotherwith large catapults that are made with surgical hose mounted on a window frame. A 0.5 kg balloon filled withdyed water is placed in a pouch attached to the hose, which is then stretched through the width of the room.Assume that the stretching of the hose obeys Hooke’s law and has a spring constant of 100 N/m.
Question
Assume that the hose is stretched by 5.00 m and then released. What is the speed of the balloon in the pouchwhen the hose reaches its relaxed length?
Rotational Motion of Solid Objects
Q C Q
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Scenario
During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one anotherwith large catapults that are made with surgical hose mounted on a window frame. A 0.5 kg balloon filled withdyed water is placed in a pouch attached to the hose, which is then stretched through the width of the room.Assume that the stretching of the hose obeys Hooke’s law and has a spring constant of 100 N/m.
Question
Assume that the hose is stretched by 5.00 m and then released. What is the speed of the balloon in the pouchwhen the hose reaches its relaxed length?
Answer
To answer this question, we will use conservation of energy
Rotational Motion of Solid Objects
A R i Q i i i R i l M i C f M T Fi l Q i
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Scenario
During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one anotherwith large catapults that are made with surgical hose mounted on a window frame. A 0.5 kg balloon filled withdyed water is placed in a pouch attached to the hose, which is then stretched through the width of the room.Assume that the stretching of the hose obeys Hooke’s law and has a spring constant of 100 N/m.
Question
Assume that the hose is stretched by 5.00 m and then released. What is the speed of the balloon in the pouchwhen the hose reaches its relaxed length?
Answer
To answer this question, we will use conservation of energy
When the hose is stretched, the energy of the system is all in the form of elastic potential energy
E i = 12 kx
2
Rotational Motion of Solid Objects
A t R i Q titi i R t ti l M ti C t f M T Fi l Q ti
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Scenario
During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one anotherwith large catapults that are made with surgical hose mounted on a window frame. A 0.5 kg balloon filled withdyed water is placed in a pouch attached to the hose, which is then stretched through the width of the room.Assume that the stretching of the hose obeys Hooke’s law and has a spring constant of 100 N/m.
Question
Assume that the hose is stretched by 5.00 m and then released. What is the speed of the balloon in the pouchwhen the hose reaches its relaxed length?
Answer
To answer this question, we will use conservation of energy
When the hose is stretched, the energy of the system is all in the form of elastic potential energy
E i = 12 kx
2
When the hose reaches its relaxed length, the energy of the system is all in the form of kinetic energy of
the balloon E f = 12 mv
2
Rotational Motion of Solid Objects
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Announcements Review Quantities in Rotational Motion Center of Mass Torque Final Questions
Scenario
During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one anotherwith large catapults that are made with surgical hose mounted on a window frame. A 0.5 kg balloon filled withdyed water is placed in a pouch attached to the hose, which is then stretched through the width of the room.
Assume that the stretching of the hose obeys Hooke’s law and has a spring constant of 100 N/m.
Question
Assume that the hose is stretched by 5.00 m and then released. What is the speed of the balloon in the pouchwhen the hose reaches its relaxed length?
Answer
To answer this question, we will use conservation of energy
When the hose is stretched, the energy of the system is all in the form of elastic potential energy
E i = 12 kx
2
When the hose reaches its relaxed length, the energy of the system is all in the form of kinetic energy of
the balloon E f = 12 mv
2
By conservation of energy, E i = E f ; solving for v :
v =
kx 2
m
1/2
=
(100 N/m)(5.00 m)2
0.5 kg
1/2
= 70.7 m/s
Rotational Motion of Solid Objects
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Quantities in translational motion
Mass (scalar): a measure of an object’s inertia
Position (vector): an object’s location
Velocity (vector): change in an object’sposition with time
Speed (scalar): the distance an object travelsin some amount of time
speed =distance
time
Acceleration (vector): change in an object’svelocity with time
Force (vector): a “push” or a “pull”
Units in translational motion
Mass: kilogram (kg)Position: meter (m)
Velocity: meter-per-second (m/s)
Acceleration: meter-per-second2 (m/s2)
Force: newton (1 N = 1 kg · m/s2 )
Rotational Motion of Solid Objects
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Quantities in translational motion
Mass (scalar): a measure of an object’s inertia
Position (vector): an object’s location
Velocity (vector): change in an object’sposition with time
Speed (scalar): the distance an object travelsin some amount of time
speed =distance
time
Acceleration (vector): change in an object’svelocity with time
Force (vector): a “push” or a “pull”
Units in translational motion
Mass: kilogram (kg)Position: meter (m)
Velocity: meter-per-second (m/s)
Acceleration: meter-per-second2 (m/s2)
Force: newton (1 N = 1 kg · m/s2 )
Quantities in rotational motion
Rotational mass: a measure of an object’s rotational inertia
Angular position: an object’s orientation
Angular velocity: change in an object’s angular positionwith time
Angular speed: the angle an object rotates in some amountof time
angular speed =change in angle
time
Angular acceleration: change in an object’s angular velocitywith time
Torque (vector): a “twist” or a “spin”
Units in rotational motion
Rotational mass: kilogram-meter2
(kg · m2
)Angular position: radian (1)
Angular velocity: radian-per-second (1/s)
Angular acceleration: radian-per-second2 (1/s2 )
Torque: newton-meter (N · m)
Rotational Motion of Solid Objects
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Angular position θ
The angular position of an object is its orientationwith respect to some reference
Angular position is measured in radians (1)
What is a radian?2π radians = 360◦ so 1 radian = 57.3◦
To measure an object’s angular position we need:
A reference (“horizontal orientation”)The angle of rotationAn axis of rotation
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Q q Q
Angular velocity ω
The angular velocity is a measure of the change inan object’s angular position with time
Angular velocity is measured in radians-per-second(1/s)
Angular velocity is a vector
To determine its direction, use the right-hand rule
Curl the fingers of your right hand in thedirection of the rotationThe direction that your thumb points isthe direction of the angular velocity
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Scenario
A disk (like a record) is rotating counterclockwise (viewed from the top) at a constant angular speed of one
revolution every 5 seconds.
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Scenario
A disk (like a record) is rotating counterclockwise (viewed from the top) at a constant angular speed of one
revolution every 5 seconds.
Question #1
In which direction is the disk’s angular velocity?
Answer
Using the right-hand rule, we find that the disk’s angular velocity is directed upward
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Scenario
A disk (like a record) is rotating counterclockwise (viewed from the top) at a constant angular speed of one
revolution every 5 seconds.
Question #1
In which direction is the disk’s angular velocity?
Answer
Using the right-hand rule, we find that the disk’s angular velocity is directed upward
Question #2
What is the magnitude of the disk’s angular velocity?
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Scenario
A disk (like a record) is rotating counterclockwise (viewed from the top) at a constant angular speed of one
revolution every 5 seconds.
Question #1
In which direction is the disk’s angular velocity?
Answer
Using the right-hand rule, we find that the disk’s angular velocity is directed upward
Question #2
What is the magnitude of the disk’s angular velocity?
Answer
Since the disk is rotating 1 revolution every 5 seconds, the magnitude of its angular velocity is
ω =1 rev
5 s=
2π rad
5 s= 1.26 rad/s.
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Translational motion
Translational motion is the motion of an object from one place to another (what we have discussed so far)
Rotational motion
Rotational motion is the motion of an object around a point
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Translational motion
Translational motion is the motion of an object from one place to another (what we have discussed so far)
Rotational motion
Rotational motion is the motion of an object around a point
Newton’s First Law (Law of Inertia)
“An object will stay at rest or continue at a constant velocity unless acted upon by an external unbalanced force”
Newton’s First Law of Rotational Motion (Law of Rotational Inertia)“A rigid object (that is not wobbling) will stay at rest or continue rotating at a constant angular velocity unlessacted upon by an external unbalanced torque”
A body at rest tends to remain at rest
A body that’s rotating tends to remain rotating
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Translational motion
Translational motion is the motion of an object from one place to another (what we have discussed so far)
Rotational motion
Rotational motion is the motion of an object around a point
Newton’s First Law (Law of Inertia)
“An object will stay at rest or continue at a constant velocity unless acted upon by an external unbalanced force”
Newton’s First Law of Rotational Motion (Law of Rotational Inertia)“A rigid object (that is not wobbling) will stay at rest or continue rotating at a constant angular velocity unlessacted upon by an external unbalanced torque”
A body at rest tends to remain at rest
A body that’s rotating tends to remain rotating
Physics Concept: Rotational inertia
Rotational inertia: the resistance of an object to a change in its rotation
Physics Concept: Rotational mass
Rotational mass (moment of inertia): the measure of an object’s rotational inertia
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Rotational mass (moment of inertia)
Some objects are easier to spin than others
The rotational mass or moment of inertia of an object is a measure of its rotational inertia (the resistanceto change in angular velocity)
The rotational mass of an object depends on:
the distribution of mass in the object
the axis about which the object rotates
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Rotational mass (moment of inertia)
Some objects are easier to spin than others
The rotational mass or moment of inertia of an object is a measure of its rotational inertia (the resistanceto change in angular velocity)
The rotational mass of an object depends on:
the distribution of mass in the object
the axis about which the object rotates
How it works
The farther away the object’s mass is from the axis of rotation, the larger its rotational mass
An object has a different rotational mass for every possible axis of rotation
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Why is one rod easier to rotate?
Both rods have the same total mass (andweight) and dimensions
Each rod is rotated about its center of mass
The red colored rod is easy to rotateThe blue colored rod is difficult torotate
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Why is one rod easier to rotate?
Both rods have the same total mass (and
weight) and dimensions
Each rod is rotated about its center of mass
The red colored rod is easy to rotateThe blue colored rod is difficult torotate
The difference between the two rods is inhow their mass is distributed
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Why is one rod easier to rotate?
Both rods have the same total mass (and
weight) and dimensions
Each rod is rotated about its center of mass
The red colored rod is easy to rotateThe blue colored rod is difficult torotate
The difference between the two rods is inhow their mass is distributed
The mass of the red rod is locatedat the point of rotationThe mass of the blue rod is locatedat the two ends
The blue rod has a large rotational masswhen rotating about its midpoint because itsmass is located far from this point
The red rod has a small rotational masswhen rotating about its midpoint because itsmass is located at this point
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Question
In 1974, Frenchman Philippe Petit, walked (and danced) on a cable suspended between the World Trade Centertowers. He carried with him a custom-made 8.0 m pole. Why?
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Question
In 1974, Frenchman Philippe Petit, walked (and danced) on a cable suspended between the World Trade Centertowers. He carried with him a custom-made 8.0 m pole. Why?
AnswerThe pole increased his rotational inertia, thereby increasing his resistance to rotate (which would cause him to fall)
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Physics Concept: Center of mass
The center of mass of an object is the point about which an object’s mass balances
Properties of the center of mass
An object behaves as if all of its mass is at its center of mass (we can imagine that the gravitational forceis only acting at the object’s center of mass)
A freely rotating object (one without a fixed axis) rotates about its center of mass
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Physics Concept: Center of mass
The center of mass of an object is the point about which an object’s mass balances
Properties of the center of mass
An object behaves as if all of its mass is at its center of mass (we can imagine that the gravitational forceis only acting at the object’s center of mass)
A freely rotating object (one without a fixed axis) rotates about its center of mass
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Example: diving
Though the motion of a diver through the air mayappear complicated, the trajectory of the diver isactually quite simple
While the diver is in the air, she rotates freelyabout her center of mass
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Example: diving
Though the motion of a diver through the air mayappear complicated, the trajectory of the diver isactually quite simple
While the diver is in the air, she rotates freelyabout her center of mass
The diver’s center of mass behaves exactly likethrown objects we have already studied (itstrajectory is a parabola)
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Example: diving
Though the motion of a diver through the air mayappear complicated, the trajectory of the diver isactually quite simple
While the diver is in the air, she rotates freelyabout her center of mass
The diver’s center of mass behaves exactly likethrown objects we have already studied (itstrajectory is a parabola)
In a single jump, how does the diver sometimesspin faster than other times?
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Example: diving
Though the motion of a diver through the air mayappear complicated, the trajectory of the diver isactually quite simple
While the diver is in the air, she rotates freelyabout her center of mass
The diver’s center of mass behaves exactly likethrown objects we have already studied (itstrajectory is a parabola)
In a single jump, how does the diver sometimesspin faster than other times? (We will learn theanswer to that question in the next lecture!)
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Question
In the 1968 Olympics, Dick Fosbury introduced a new style of jumping to the high-jump event and won the goldmedal. Why is the peculiar form of the so-called Fosbury advantageous?
Answer
It lowers the altitude of the center of mass of the athlete
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Newton’s First Law (Law of Inertia)
“An object will stay at rest or continue at a constant velocity unless acted upon by an external unbalanced force.”
Newton’s First Law of Rotational Motion (Law of Rotational Inertia)“A rigid object (that is not wobbling) will stay are rest or continue rotating at a constant angular velocity unlessacted upon by an external unbalanced torque.”
A body at rest tends to remain at rest
A body that’s rotating tends to continue rotating
Newton’s Second Law of Motion
“The net force on an object is equal to the mass m of the object multiplied by its acceleration −→a . Theacceleration is in the same direction as the net force.”
−→
F = m−→a
Newton’s Second Law of Rotational Motion
“The net torque on an object (that is not wobbling) is equal to the rotational mass I of the object multiplied by itsangular acceleration −→α . The angular acceleration is in the same direction as the net torque.”
−→τ = I −→α
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Torque
To get something to start spinning, we must apply atorque
To apply a torque we need
a pivot pointa lever arman applied force
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Torque
To get something to start spinning, we must apply atorque
To apply a torque we need
a pivot pointa lever arman applied force
The magnitude of the torque = lever arm · forceperpendicular to lever arm
τ = r · F ⊥
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Torque
To get something to start spinning, we must apply atorque
To apply a torque we need
a pivot pointa lever arman applied force
The magnitude of the torque = lever arm · forceperpendicular to lever arm
τ = r · F ⊥
The direction of the torque is given by the right-handrule:
point your right hand in the direction of thelever armcurl your fingers in the direction of the appliedforceThe direction of your outstretched thumb is the
direction of the torque
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Question (iclicker)
Two mechanics are using wrenches to loosen screws on a ship. One of the mechanics is rather wimpy and can onlyexert half the force that the other mechanic can in turning a wrench. Though weak, he is very crafty and hasdevised a custom wrench for himself that is 1 m long (the other mechanic’s wrench is only 0.5 m long). Assumingboth mechanics have perfect “wrench form”, which mechanic has an easier time loosening screws?
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Announcements Review Quantities in Rotational Motion Center of Mass Torque Final Questions
8/11/2019 Rotational Motion of Solid Objects 8.1-8.3
http://slidepdf.com/reader/full/rotational-motion-of-solid-objects-81-83 45/46
Question (iclicker)
Two mechanics are using wrenches to loosen screws on a ship. One of the mechanics is rather wimpy and can onlyexert half the force that the other mechanic can in turning a wrench. Though weak, he is very crafty and hasdevised a custom wrench for himself that is 1 m long (the other mechanic’s wrench is only 0.5 m long). Assumingboth mechanics have perfect “wrench form”, which mechanic has an easier time loosening screws?
AnswerThey are both equally as easy
Rotational Motion of Solid Objects
Announcements Review Quantities in Rotational Motion Center of Mass Torque Final Questions
8/11/2019 Rotational Motion of Solid Objects 8.1-8.3
http://slidepdf.com/reader/full/rotational-motion-of-solid-objects-81-83 46/46
Reading Assignment
Read sections 8.4 - 8.5
Homework Assignment 4
Homework for Chapter 5 is due in class today
Homework Assignment 5
Homework for Chapters 6 and 8 (due Tuesday, October 5)Chapter 6: Q6, Q13, Q24, Q32, E10, E14Chapter 8: Q6, Q12, Q26, E8, E12, E18
Rotational Motion of Solid Objects