Calculate the angular momentum of this particle in rotational motion

In summary, the conversation discusses the use of vectors i, j, and k to define angular momentum, with a reminder to use the vector valued cross product instead of regular multiplication. The anti-symmetry of the cross product is also mentioned, followed by a clarification on the cross product in two dimensions.
  • #1
YanZhen
10
0
Homework Statement
The mass of the particle is m,the equation of motion is r=acosωti+bsinωtj.a and b are constant.Calculate the angular momentum of a particle at any time.
Relevant Equations
L=P*r=m*v*r
v=dr/dt
i,j,k arevector
I know L=P*r=m*v*r=m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k=mabωk.
but why m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k.I need some detail.
please help me.
 
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  • #2
You should not be using regular multiplication to multiply vectors, you need to use the vector valued cross product to define the angular momentum:
$$
\vec L = \vec p \times \vec r
$$
where
$$
\hat i \times \hat j = \hat k, \quad \hat j \times \hat k = \hat i, \quad \hat k \times \hat i = \hat j
$$
and furthermore you have anti-symmetry so ##\vec u \times \vec v = - \vec v \times \vec u## for any vectors ##\vec u## and ##\vec v##.
 
  • #3
IIRC $$\vec L\equiv \vec r \times\vec p$$ :wink:
 
  • #4
BvU said:
IIRC $$\vec L\equiv \vec r \times\vec p$$ :wink:
Too busy arguing about the need to use the cross product to care about getting the right definition I suppose ... 🤷‍♂️
 
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  • #5
Orodruin said:
You should not be using regular multiplication to multiply vectors, you need to use the vector valued cross product to define the angular momentum:
$$
\vec L = \vec p \times \vec r
$$
where
$$
\hat i \times \hat j = \hat k, \quad \hat j \times \hat k = \hat i, \quad \hat k \times \hat i = \hat j
$$
and furthermore you have anti-symmetry so ##\vec u \times \vec v = - \vec v \times \vec u## for any vectors ##\vec u## and ##\vec v##.
wow,thanks for your wonderful answer,it solved my question.and it also requires a point of knowlege about vectors.(x1,y1)X(x2,y2)=x1y2-y1x2
 
  • #6
YanZhen said:
(x1,y1)X(x2,y2)=x1y2-y1x2
Careful. This is not the cross product. It is a component (the third) of the cross product between two vectors that only have i and j components. The cross product as such does not exist in two dimensions (although there are generalisations of it that do - among them the wedge product, which would look something like what you wrote but is generally too advanced for basic introduction to vectors and their application). The general form of the cross product (in three dimensions) would read
$$
(x_1,y_1,z_1) \times (x_2, y_2, z_2) = (y_1 z_2 - y_2 z_1, z_1 x_2 - z_2 x_1, x_1 y_2 - x_2 y_1)
$$
 
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FAQ: Calculate the angular momentum of this particle in rotational motion

1. How do you calculate the angular momentum of a particle in rotational motion?

The angular momentum of a particle in rotational motion can be calculated by multiplying the moment of inertia of the particle by its angular velocity. The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

2. What is the moment of inertia?

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is calculated by multiplying the mass of the object by the square of its distance from the axis of rotation. The formula for moment of inertia is I = mr^2, where I is the moment of inertia, m is the mass of the object, and r is the distance from the axis of rotation.

3. How does angular velocity affect the angular momentum of a particle?

Angular velocity is directly proportional to the angular momentum of a particle. This means that as the angular velocity increases, the angular momentum also increases. This relationship is represented by the formula L = Iω, where L is the angular momentum and ω is the angular velocity.

4. Can the angular momentum of a particle change?

Yes, the angular momentum of a particle can change if there is a change in its angular velocity or moment of inertia. In the absence of external forces, the angular momentum of a particle will remain constant due to the law of conservation of angular momentum.

5. What are the units of angular momentum?

The units of angular momentum are kilogram meters squared per second (kg·m^2/s) in the SI system. In the CGS system, the units are gram centimeters squared per second (g·cm^2/s). These units can also be expressed as newton meters per second (N·m/s) or joule seconds (J·s).

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