department of physics and applied physics 95.141, f2010, lecture 20 physics i 95.141 lecture 20...

Department of Physics and Applied Physics95.141, F2010, Lecture 20

Physics I95.141

LECTURE 2011/21/10


Parallel Axis Theorem

• Moment of inertia for a sphere, rotating about an axis through its center, is:

• What is moment of inertia for this sphere about an axis going through the edge of the sphere?

• Through an axis a distance R from the center?

• As R>>ro?

R


Rotational Kinetic Energy

• We now know the rotational equivalent of mass is the moment of inertia I.

• If I told you there was such a thing as rotational kinetic energy, you could probably make a good guess as to what form it would take…

2

2

1mvKEtrans

2

2

1 IKErot


Rotational Kinetic Energy

• Prove it!

2

2

1mvKE trans Rv


Rolling without slipping

• If the CM of the wheel is moving with speed v

• Place yourself at the axis, you will see road moving by at –v beneath you.

• So if the road is moving at –v, and the wheel isn’t slipping, you know the angular velocity of the wheel!

Rv

Rv

/


Instantaneous Axis

• What is the axis of rotation for the rolling wheel?• The point on the ground is fixed, the wheel is

then rotating around this point.

2MRII CM


Total KE of Rolling Body

• The total kinetic energy of the rolling body will then be:

22

2

1 MRIKE CM

RvCM


Rolling Body Example


• What is speed of rock at end of incline?• Use conservation of Energy

Rolling Body Example

2

53

5

2

5

1076.13

4

5.2

MRI

my

kgRMrock

mR

sphere

ROCK


Outline• Angular Momentum

• Vector Cross Products

• What do we know?

– Units

– Kinematic equations

– Freely falling objects

– Vectors

– Kinematics + Vectors = Vector Kinematics

– Relative motion

– Projectile motion

– Uniform circular motion

– Newton’s Laws

– Force of Gravity/Normal Force

– Free Body Diagrams

– Problem solving

– Uniform Circular Motion

– Newton’s Law of Universal Gravitation

– Weightlessness– Kepler’s Laws– Work by Constant Force– Scalar Product of Vectors– Work done by varying Force– Work-Energy Theorem– Conservative, non-conservative Forces– Potential Energy– Mechanical Energy – Conservation of Energy– Dissipative Forces– Gravitational Potential Revisited– Power– Momentum and Force– Conservation of Momentum– Collisions– Impulse– Conservation of Momentum and Energy– Elastic and Inelastic Collisions2D, 3D Collisions– Center of Mass and translational motion– Angular quantities– Vector nature of angular quantities– Constant angular acceleration – Torque– Rotational Inertia– Moments of Inertia


Review of Lecture 19

• Introduced concept of Torque– “Rotational Force”

• Comes from Force exerted on a rotating body• Depends on distance from axis of rotation• Equal to radius x perpendicular component of Force• Does not have same units as Force

• Rotational equivalent of mass is moment of inertia…how hard it is to accelerate (angularly) an object for a given Torque.– For a point mass: – For a collection of point masses:– For solid object rotating around CM: use table in book, or– use integral:

][ mNRF

2MRI ...2

33222

211 RMRMRMI

dmrI 2


Lecture 19 Review

• Parallel axis theorem– Can be used to determine I around an axis other than one

through center of mass

• Discussed equivalent of Newton’s 2nd Law for Angular Motion– Torque is what gives angular acceleration:

• Angular Kinetic Energy

• Rolling without slipping

IRF

2

2

1 IKEangular

22

2

1 MRIKE CM RvCM


What are we missing?

• We have discussed angular acceleration, velocity, Force (torque) and kinetic energy

• All have equivalents in linear motion• There is one other translational quantity we have

spent a lot of time with…momentum, does this have an angular equivalent?

• Again, if we had to guess….


Angular Momentum

• Angular momentum is defined as the product angular velocity of an object and the object’s moment of inertia.

• Can write the equivalent of Newton’s second law for angular momentum, just like we did for translational momentum


Conservation of Angular Momentum

• Just like translational momentum, angular momentum is a conserved quantity in certain circumstances.

• Because a torque leads to angular acceleration (a change in angular velocity), if a torque is applied to a rotating body, angular momentum is not conserved.

• But if no external torques act on a body, and I is constant, then angular momentum is conserved!


Conservation of Angular Momentum

• Law of conservation of angular momentum:• The total angular momentum of a rotating object

remains constant if the net external torque acting on the object is zero

• Import conservation law in Physics, along with conservation of linear momentum and energy!!


Example Problem

• Lets take the rotating rod system we looked at earlier.• If the rod is rotating at ω=5rad/s, what is the system’s angular

momentum?

• If I remove block 2, what is the new angular velocity of the system?

m2=3kg m1=2kg2m2m


Directional Nature of L

• For translational momentum, the momentum vector points in the same direction as the velocity vector, because mass is a scalar.

• In the same way, because the moment of inertia is a scalar, the angular velocity vector determines the direction of the angular momentum vector.


Vector Cross Product

• We have discussed, so far, two types of vector products:– Multiplication by a scalar:

– Dot-product of two vectors ( )

kjiAc ˆ2ˆ4ˆ3,2

Ac

dF

kjiBkjiA ˆ4ˆ5ˆ2,ˆ2ˆ4ˆ3

BA


Vector Cross-Product

• There is yet a third type of vector product, known as the vector cross product

• Product of two vector, that results in a vector

• Resulting vector has a magnitude

• And points in the direction perpendicular to the plane created by• Direction determined by right hand rule

BAC

sinABBAC

BA,


Vector Cross Product, Mathematically

zyx

zyx

zyx

zyx

BBB

AAA

kji

BA

kBjBiBB

kAjAiAA

ˆˆˆ

ˆˆˆ

ˆˆˆ


Vector Cross Product

• What is the vector cross product of a vector and itself?

zyx

zyx

zyx

AAA

AAA

kji

AA

kAjAiAA

ˆˆˆ

ˆˆˆ


Does Order Matter?

zyx

zyx

zyx

zyx

zyx

zyx

AAA

BBB

kji

AB

BBB

AAA

kji

BA

kBjBiBB

kAjAiAA

ˆˆˆ

ˆˆˆ

ˆˆˆ

ˆˆˆ


Example

• What are the vector cross products

ji ˆˆ kj ˆˆ ii ˆˆ


Example

• What is the vector dot product of the two vectors:

kjiB

kjiA

ˆ1ˆ2ˆ1

ˆ4ˆ3ˆ2


Torque and the Cross Product

• Technically, when we first introduced torque as the product of the radius and the perpendicular component of the Force, we were taking a cross product!

FR


Angular Momentum of a particle

• Imagine a particle of mass m moving in a circular path of radius R with angular velocity ω.


Angular Momentum of Particle

• If we think about vectors, it becomes clear that the angular momentum must be the cross product of the linear momentum and the radius!

department of physics and applied physics 95.141, f2010, lecture 20 physics i 95.141 lecture 20...

Documents

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