Rotational Motion of Solid Objects 8.1-8.3

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


http://find/

http://goback/




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


http://find/




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


http://find/




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.


http://find/




Scenario


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?


Q C Q

http://find/




Scenario


Question


Answer

To answer this question, we will use conservation of energy


A R i Q i i i R i l M i C f M T Fi l Q i

http://find/




Scenario


Question


Answer


When the hose is stretched, the energy of the system is all in the form of elastic potential energy

E i = 12 kx

2


A t R i Q titi i R t ti l M ti C t f M T Fi l Q ti

http://find/




Scenario


Question


Answer



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



http://find/




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


Answer



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



http://find/




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 )



http://find/

http://goback/




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)



http://find/

http://goback/




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



http://find/

http://goback/



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



http://find/



Scenario

A disk (like a record) is rotating counterclockwise (viewed from the top) at a constant angular speed of one

revolution every 5 seconds.


http://find/



Scenario



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



http://find/



Scenario



Question #1


Answer


Question #2

What is the magnitude of the disk’s angular velocity?



http://find/



Scenario



Question #1


Answer


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.



http://find/



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



http://find/





Rotational motion


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



http://find/





Rotational motion



“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 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



http://find/



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



http://find/



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



http://find/



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



http://find/




Both rods have the same total mass (and

weight) and dimensions



The difference between the two rods is inhow their mass is distributed



http://find/




Both rods have the same total mass (and

weight) and dimensions



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



http://find/



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?



http://find/



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)


http://find/

http://goback/



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



http://find/



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


http://find/



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



http://find/



Example: diving



The diver’s center of mass behaves exactly likethrown objects we have already studied (itstrajectory is a parabola)



http://find/



Example: diving




In a single jump, how does the diver sometimesspin faster than other times?



http://find/



Example: diving




In a single jump, how does the diver sometimesspin faster than other times? (We will learn theanswer to that question in the next lecture!)


http://find/



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



http://find/




“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 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



http://find/



Torque

To get something to start spinning, we must apply atorque

To apply a torque we need

a pivot pointa lever arman applied force



http://find/



Torque




The magnitude of the torque = lever arm · forceperpendicular to lever arm

τ = r · F ⊥



http://find/



Torque




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


http://find/



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?



http://find/



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



http://find/



Reading Assignment

Read sections 8.4 - 8.5


Homework for Chapter 5 is due in class today


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


http://find/

Rotational Motion of Solid Objects 8.1-8.3

Documents

Transcript of Rotational Motion of Solid Objects 8.1-8.3