Download Review Notes on Angular Momentum, Correspondence Between

AP Physics C
Mrs. Coyle
Review Notes on:
 Angular Momentum and its Conservation
 Correspondence Between Translational and Linear Quantities
 Rolling
Angular Momentum
1) Angular Momentum of a particle about the origin:
⃑ = ⃑⃑𝒓 𝒙 𝒑
⃑
𝑳
⃑𝑳 = ⃑⃑𝒓 𝒙 𝒎𝒗
⃑⃑⃑ = r m v sin θ
where 𝑟 is the position vector with respect to the origin,
𝑣 is the velocity of the particle and θ is the angle between the r and the v vectors when
they are tail to tail .
The direction of the angular momentum vector is always perpendicular to the 𝑟-𝑣 plane.
Use the right hand curl rule. Wrap fingers from the 𝑟 to the 𝑣, then the angular
momentum vector is in the direction of the thumb. Alternative way to find the direction
of angular momentum is to use the right hand curl rule and curl your fingers in the
direction of ω, then the thumb is in the direction of angular momentum.
2) Angular Momentum of a System of Particles is the vector sum of their
individual angular momenta.
3) Angular Momentum of a Rigid Body About a Fixed Axis of Rotation
L=Iω
Torque and Angular Momentum (Newton’s second a Law in angular form)
𝒅𝑳
τ=
𝒅𝒕
Conservation of Angular Momentum
In the absence of an external torque (isolated system) angular momentum is conserved
regardless of what change takes place within the system.
τ=
𝒅𝑳
𝒅𝒕
=0, L=constant
Li= Lf
Ii ωi = If ωf
Correspondence Between Translational and Rotational Motion
Translational
Rotational
Force F=ma
Linear Momentum p=mv
Newton’s Second Law
F=ma
𝒅𝒑
F=
𝒅𝒕
Conservation in an Isolated system
(absence of external force)
Torque τ= r x F
Angular Momentum L=r x mv
Newton’s Second law
τ= I α
𝒅𝑳
τ=
𝒅𝒕
Conservation in an Isolated Sytem
(absence of external torque)
p= constant
Kinetic Energy
𝟏
K= 𝟐 𝒎𝒗𝟐
Inertia
m
Work
W=∫ 𝒇 ∙ 𝒅𝒙
Power
𝒅𝑾
P= 𝒅𝒕
For constant force
P= Fv
L=constant
Rotational Kinetic Energy
𝟏
K= 𝟐 𝑰𝝎𝟐
Moment of Inertia
I= ∫ 𝒓𝟐 𝒅𝒎
Work
W=∫ 𝝉 ∙ 𝒅𝜽
Power
For constant torque
P= τω
Rolling Object
Rolling is a combination of Translation and Rotation.
Characteristics:
Linear speed of the center of mass:
vcom = ω r
where ω is the angular speed of the wheel about its center
Linear speed of a point at the top of the wheel v=2vcom
Linear speed of the point touching the “road” =0
Note: the angular speed of the top of the wheel about the point touching the
road is also ω.
Kinetic Energy for Rolling K= ½ mv2 + ½ I ω2