Download L11_torque

Warm-up
How does an object’s momentum change if
it receives a net force perpendicular to its
velocity?
v
F
How about its kinetic energy?
Announcement
• Lab this week leads class. That’s on
purpose.
• There will be a rotation lab next week.
Torque and Angular
Momentum
Vectors of rotation
Objectives
• Determine the torque applied by a force
about an axis.
• Predict an object’s response to a torque.
• Calculate an object’s angular momentum.
Poll Question
You push on a door. It will open easiest if you
push
A. opposite the hinge.
B. at the center of the
door.
C. near the hinge.
Torque
Turning force = torque
= radius  force
=rF
Units: Nm (not J)
Vector Cross Product
•
•
•
•
Operation symbol 
Another way to multiply two vectors
Product is a vector!
Direction of AB is perpendicular to
both A and B
Cross Product Magnitude
a
q
A  B = ab sin q
A
B
b
Maximum for q = 90°
Zero for q = 0°, 180°
Magnitude Geometrically
a
q
A
B
b
AB = area of parallelogram
Cross Product Direction
a
q
A
B
b
• Curl right-hand fingers in
direction of q
• Right-hand thumb points
in direction of crossproduct
• Not commutative
AB = –(BA)
Radius Vector
• Radius from reference point to application
of force
• Strictly, reference point must be specified
to determine a torque (about the point)
• Torque depends on your choice of point!
Adding Torques
750 N
• Net torque about fulcrum is zero here
• Torques are vectors
Whiteboard Work
A 10,000-N truck is stalled 1/4 of the way across a
100-m bridge. What torque does its weight apply
about the far (right) support?
r
Whiteboard Work
What upward force must the near (left) support
provide to cancel the truck’s torque about the far
support?
r
F
Whiteboard Work
What upward force must the far support provide to
support the weight of the truck?
Hint: Several ways will work:
• canceling forces on the truck
• canceling torques about the near support
r
F
Angular Momentum
• Torque is rotational force
• Angular momentum is rotational
momentum
L=rp
• Angular momentum is a vector
• Direction by right-hand rule
Poll Question
The moon revolves around the Earth with a period
of 27.322 days. If the moon were to move farther
away from the Earth but maintain its angular
momentum, how would its period adjust?
A.
B.
C.
Its period would become shorter than 27.322 days.
Its period would become longer than 27.322 days.
Its period would remain 27.322 days.
Newton’s Second Law
• Force is the rate of change of momentum
Dp
F=
Dt
• Torque is the rate of change of angular
momentum
DL
t=
Dt
Conservation of Angular
Momentum
• If no outside torque, l = r  p is constant.
• If r decreases, p increases!
Conservation of Angular
Momentum
• Nothing can apply a torque to itself.
• Any change in one object’s angular
momentum is accompanied by an
opposite change in another object.
• The angular momentum of the universe
never changes.
Conservation of Momentum
Q. How can linear momentum be conserved
if p increases?
A. Total linear momentum is zero in a
rotating system!
Conservation of Energy
Q. What happens to kinetic energy when p
increases?
A. Kinetic energy increases! DE = w.
Work is done to pull rotating parts inward.
Gyroscopic Stability
More angular momentum
Objectives
• Determine the torque applied by a force
about an axis.
• Predict an object’s response to a torque.
• Calculate an object’s angular momentum.
Angular Momentum
• Angular momentum is rotational
momentum
L=rp
• Torque is the rate of change of angular
momentum
DL
t=
Dt
Gyroscopic Stability
Q. What happens when you apply a torque
to an object in a direction different from its
rotation?
A. Its axis of rotation rotates!
Adding Angular Momentum
L0
force
DL
L0 + DL = L
l0
DL
L
L
Reading for Next Time
• Conservation laws
– They tell us about the nature of the Universe
• Elasticity
– A simple but surprisingly useful model of solid
behavior