Transcript Rotational Motion

Rotational Vectors and
Angular Momentum
Angular Velocity
• Angular velocity is a vector and its direction
is perpendicular to the plane of rotation.
• Right-hand Rule: Curl your fingers in the
direction of rotation, then the thumb points
in the direction of angular velocity.
• Counter-clockwise rotation - Positive
• Clockwise rotation - Negative
Angular
Acceleration

d

dt

• What is the average angular acceleration for
a top when the angular velocity goes from
4iˆ  16kˆ rad / s to 4 ˆj  16kˆ rad / s in 0.5s?




z
z
y
x
y
x
Torque
•
•
•
•
Magnitude -   rF sin 
Direction - Use the right hand rule.



Mathematically -   r  F
Matrix Method - Find the determinant of the
following matrix.
ˆ
iˆ

  rx
Fx
ˆj
ry
Fy
k
rz
Fz
Torque Example
• What is the torque exerted about the origin
by a 5iˆ  3kˆ N force if it is applied at the
point 4 ˆj  2kˆ m ?




Angular Momentum
• For a particle, angular momentum is
calculated by the formula
  
Lrp
• Ex. What is the angular momentum of the
Halley’s Comet at perihelion?
r  8.79 10 m
3
v  54.6 10 m / s
5
m  3.0 10 kg
10
What happens to Halley’s
Comet as it swings away
from the sun?
Conservation of
Angular Momentum
• In terms of angular velocity


L  I
• Analogous rotational Newton’s 2nd Law

 dL

dt
• Conservation of Angular Momentum
– If there is no net external torque on a system,
then the angular momentum is constant.
Merry-Go-Round
• A merry-go-round is similar to a circular
disk with a mass of 100kg and a radius of
2.0m. It is rotating at 1.57 rad/s when a
stationary man of mass 70kg jumps on the
edge. What is the new angular velocity?
• The man now moves to within 1.0m of the
center. Now what is the angular velocity
Conservation of
Angular Momentum
• Orientation of earth is maintained
• Plane of earth’s orbit is maintained.
• Ex. Gyroscope
N.P.
Sun
Eq.
S.P.