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

• What is the average angular acceleration for a top when the angular velocity goes from

to in 0.5s?

dt

d

sradki / ˆ16ˆ4 sradkj / ˆ16ˆ4 z

x

y

z

x

y

Torque• Magnitude -

• Direction - Use the right hand rule.

• Mathematically -

• Matrix Method - Find the determinant of the following matrix.

sinrF

Fr

zyx

zyx

FFF

rrr

kji ˆˆˆ

Torque Example• What is the torque exerted about the origin

by a force if it is applied at the point ?

Nki ˆ3ˆ5 mkj ˆ2ˆ4

Angular Momentum

• For a particle, angular momentum is calculated by the formula

• Ex. What is the angular momentum of the Halley’s Comet at perihelion?

prL

mr 101079.8 smv /106.54 3

kgm 5100.3

What happens to Halley’sComet as it swings awayfrom the sun?

Conservation of Angular Momentum

• In terms of angular velocity

• Analogous rotational Newton’s 2nd Law

IL

dt

Ld

• 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

Eq.

N.P.

Sun

S.P.