Download Angular Momentum: Understanding the Vector Product, Torque, and Conservation and more Study notes Physics in PDF only on Docsity!
Chapter 11
Angular Momentum
The Vector Product
There are instances where the product of two
vectors is another vector
Earlier we saw where the product of two vectors
was a scalar
This was called the dot product
The vector product of two vectors is also
called the cross product
The Vector Product and Torque
The torque vector lies in a direction perpendicular to the plane formed by the position vector and the force vector
ττττ = r x F The torque is the vector (or cross) product of the position vector and the force vector
The Vector Product Defined
Given two vectors, A and B
The vector (cross) product of A and B is
defined as a third vector, C
C is read as “ A cross B ”
The magnitude of C is AB sin θ
θ is the angle between A and B
More About the Vector Product
The quantity AB sin θ is equal to the area of the parallelogram formed by A and B The direction of C is perpendicular to the plane formed by A and B The best way to determine this direction is to use the right-hand rule
Properties of the Vector Product
The vector product is not commutative. The
order in which the vectors are multiplied is
important
To account for order, remember
A x B = - B x A
If A is parallel to B (θ = 0o^ or 180o), then A x
B = 0
Therefore A x A = 0
Vector Products of Unit
Vectors
ˆ ˆ^ ˆ^ ˆ ˆ
× = × = × =
× = − × =
× = − × =
× = − × =
i i j j k k
i j j i k
j k k j i
k i i k j
Vector Products of Unit
Vectors, cont
Signs are interchangeable in cross products
A x (- B ) = - A x B
ˆi × ( −ˆj) =−ˆi׈j
Torque and Angular
Momentum
The torque is related to the angular momentum
Similar to the way force is related to linear momentum
This is the rotational analog of Newton’s Second
Law
and L must be measured about the same origin This is valid for any origin fixed in an inertial frame
d
dt
∑^ τ =
d ( ) L
dt
×
∑ =
r p
Σ τ
More About Angular
Momentum
The SI units of angular momentum are r x p =
(kg.m^2 )/ s
Both the magnitude and direction of L
depend on the choice of origin
The magnitude of L = mvr sin φ
φ is the angle between p and r
The direction of L is perpendicular to the
plane formed by r and p
Angular Momentum of a
Particle, Example
The vector L = r x p is pointed out of the diagram
The magnitude is L = mvr sin 90o^ = mvr sin 90o^ is used since v is perpendicular to r A particle in uniform circular motion has a constant angular momentum about an axis through the center of its path
Angular Momentum of a
Rotating Rigid Object
Each particle of the object rotates in the xy plane about the z axis with an angular speed of ω The angular momentum of an individual particle is Li = mi ri^2 ω L and ωωωω are directed along the z axis
Angular Momentum of a
Rotating Rigid Object, cont
To find the angular momentum of the entire
object, add the angular momenta of all the
individual particles
This also gives the rotational form of
Newton’s Second Law
2 z i i i i i
L = (^) ∑ L = (^) ∑ m r ω = I ω
ext^ z
dL (^) I d I dt dt
τ = = ω=α
Angular Momentum of a
Bowling Ball
The momentum of
inertia of the ball is
2/5 MR^2
The angular
momentum of the ball
is Lz = I ω
The direction of the
angular momentum is
in the positive z
direction
Conservation of Angular
Momentum
The total angular momentum of a system is constant
in both magnitude and direction if the resultant
external torque acting on the system is zero
Net torque = 0 -> means that the system is isolated
L tot = constant or L i = L f
For a system of particles, L tot = Σ L n = constant
Conservation of Angular
Momentum, cont
If the mass of an isolated system undergoes
redistribution, the moment of inertia changes
The conservation of angular momentum requires a compensating change in the angular velocity Ii ω i = If ω f This holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system The net torque must be zero in any case
