Physics 218, Lecture XXI 1 Physics 218 Lecture 21 Dr. David Toback
Dec 20, 2015
Physics 218, Lecture XXI 1
Physics 218Lecture 21Dr. David Toback
Physics 218, Lecture XXI 2
Exam Next Tuesday
Today’s Topics notnot covered on the exam, but are a review
Physics 218, Lecture XXI 3
General Rotation
• Chapter 11• Today’s lecture will be about extending the
angular stuff we did last week.• In many ways it’s a summary
–Some math–Torque–Angular Momentum–Kepler’s 2nd Law
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Physics 218, Lecture XXI 5
Physics 218, Lecture XXI 6
Vector Cross Product
This is the last way of multiplying vectors we will see
A B vs.B A
A A
:Check
B" intoA Swing"
Rule-Hand-Right fromDirection
SinB AC
B A C
Pneumonic: Matrix
Physics 218, Lecture XXI 7
Torque Redefined
A better way to define Torque
This gives us the magnitude and the direction
FR
Physics 218, Lecture XXI 8
Example of Cross Product
The location of a body is length r from the origin and at an angle from the x-axis. A force F acts on the body purely in the y direction.
What is the Torque on the body?
z
x
y
Physics 218, Lecture XXI 9
Angular Momentum
Again we use the Cross Product:
p r
L
Physics 218, Lecture XXI 10
Angular Motion of a Particle
Determine the angular momentum, L, of a particle,with mass m and speed v, moving in uniform circular motion with radius r.
Physics 218, Lecture XXI 11
Kepler’s 2nd Law
2nd Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time
Physics 218, Lecture XXI 12
A bullet strikes a cylinderA bullet of speed V and
mass m strikes a solid cylinder of mass M and inertia ½MR2, at radius R and sticks. The cylinder is anchored at point 0 and is initially at rest.
What is of the system after the collision?
Is energy Conserved?
Physics 218, Lecture XXI 13
Atwood’s Machine
A pulley with a fixed center (at point O), radius R and moment of inertia I, has a massless rope wrapped around it (no slipping). The rope has two masses, m1 and m2 attached to its ends. Assume m2>m1
• What is the acceleration of the system?
• Do some checks.
Physics 218, Lecture XXI 14
Next Period: Exam• There is an exam here, next class period (Tuesday)• Same as usual:
– Problems from the Textbook, Homework and Lecture
– Formula sheet handed out
– All formulas you use must be derived from those on the formula sheet
• Topics are: Chapters 8-10– Conservation of Energy
– Momentum and Collisions
– Rotational Motion
• Today’s Topics notnot covered, but are a review
Physics 218, Lecture XXI 15
Next Tuesday: Exam• There is an exam here, not next class period, but Tuesday• Same as usual:
– Problems from the Textbook, Homework and Lecture– Formula sheet handed out– All formulas you use must be derived from those on the
formula sheet
• Topics are: Chapters 8-10– Conservation of Energy– Momentum and Collisions– Rotational Motion
• Today’s Topics notnot covered, but are a review • Next week’s topics: Oscillations
Physics 218, Lecture XXI 16
Why does the Bicycle Wheel Turn to the Right?
Physics 218, Lecture XXI 17
Angular Momentum
Again we use the Cross Product:
Derivation of
= dL/dt
p r
L
Physics 218, Lecture XXI 18
L for a system of many bodies
• Have to be careful with Angular Momentum– = dl/dt for a single particle
– = dl/dt) for a system of many particles
– All internal torques cancel because of Newton’s law (all internal forces are equal and opposite)
• Reference Frame matters. Only true for:– The origin is an inertial Reference Frame
– The center of mass
Physics 218, Lecture XXI 19
L for a Rigid Body
Find the angular momentum, L, for this body given that it is rotating around the Z axis with angular velocity