Q: In the second image, a force $$F is $$applied at an angle $$\theta to $$the arm. If the position vector from point $$A to $$the hand is $$\vec{r} = b \hat{x} + h \hat{y}$$, what is the moment (torque) about point $$A$$?

$$-F b \cos \theta + h F \sin \theta$$

Q: What is the angular momentum $$\vec{H}_O of a $$particle with position vector $$\vec{r}$$ and linear momentum $$\vec{p}$$ relative to point $$O$$?

$$\vec{H}_O = \vec{r} \times \vec{p}$$

Q: For a simple pendulum of length $$L$$ and mass $$m$$, what is the angular momentum about the pivot $$O if $$the angular velocity is $$\dot{\theta}$$?

$$m L^2$ \dot{\theta} \hat{z}$$

Q: What is the equation of motion for the simple pendulum using the angular momentum principle?

$$\ddot{\theta} + \frac{g}{L} \sin \theta = 0$$

Q: In cylindrical coordinates, what is the $$z-$$component of the cross product $$\vec{u} \times \vec{v}$$ for $$\vec{u} = (u_r, u_\theta, u_z)$$ and $$\vec{v} = (v_r, v_\theta, v_z)$$?

$$u_r v_\theta - u_\theta v_r$$

Q: If $$\vec{u}$$ and $$\vec{v}$$ are vectors, what is the magnitude of their cross product $$\vec{u} \times \vec{v}$$?

$$|\vec{u}| |\vec{v}| \sin \theta$$

Question 1

Which rule is illustrated in the first image, and what does it determine in vector operations?

Accepted Answer

The right-hand rule, which determines the direction of the cross product $$\vec{u} 	imes \vec{v}.$$

Question 2

Given vectors $$\vec{u} = (u_x, u_y, u_z)$$ and $$\vec{v} = (v_x, v_y, v_z) in $$Cartesian coordinates, what is the $$z-$$component of their cross product $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

Question 3

In the second image, a force $$F is $$applied at an angle $$	heta to $$the arm. If the position vector from point $$A to $$the hand is $$\vec{r} = b \hat{x} + h \hat{y}$$, what is the moment (torque) about point $$A$$?

Accepted Answer

$$-F b \cos 	heta + h F \sin 	heta$$

Question 4

Which property of the cross product is demonstrated by $$\vec{u} 	imes \vec{v} = -\vec{v} 	imes \vec{u}$$?

Accepted Answer

Anti-commutativity

Question 5

A particle of mass $$m$$ moves with velocity $$\vec{v}. $$What is its linear momentum $$\vec{p}$$?

Accepted Answer

$$\vec{p} = m \vec{v}$$

Question 6

What is the angular momentum $$\vec{H}_O of a $$particle with position vector $$\vec{r}$$ and linear momentum $$\vec{p}$$ relative to point $$O$$?

Accepted Answer

$$\vec{H}_O = \vec{r} 	imes \vec{p}$$

Question 7

If the position vector $$\vec{r}$$ and momentum $$\vec{p}$$ are parallel, what is the angular momentum $$\vec{H}_O$$?

Accepted Answer

$$\vec{H}_O is $$zero

Question 8

For a simple pendulum of length $$L$$ and mass $$m$$, what is the angular momentum about the pivot $$O if $$the angular velocity is $$\dot{	heta}$$?

Accepted Answer

$$m L^2$ \dot{	heta} \hat{z}$$

Question 9

What is the equation of motion for the simple pendulum using the angular momentum principle?

Accepted Answer

$$\ddot{	heta} + \frac{g}{L} \sin 	heta = 0$$

Question 10

Which force acting on the pendulum does NOT contribute to the moment about the pivot?

Accepted Answer

Tension

Question 11

In cylindrical coordinates, what is the $$z-$$component of the cross product $$\vec{u} 	imes \vec{v}$$ for $$\vec{u} = (u_r, u_	heta, u_z)$$ and $$\vec{v} = (v_r, v_	heta, v_z)$$?

Accepted Answer

$$u_r v_	heta - u_	heta v_r$$

Question 12

According to the principle of angular momentum, what does $$\frac{d \vec{H}_O}{dt}$$ equal?

Accepted Answer

The moment about point $$O of $$all forces acting on the particle

Question 13

If a force $$\vec{F}$$ acts along the line joining point $$O$$ and $$P$$, what is the moment $$\mathcal{M}_O \vec{F}$$ about $$O$$?

Accepted Answer

$$\mathcal{M}_O \vec{F} = 0$$

Question 14

What is the kinetic energy $$K of a $$particle with momentum $$p$$ and mass $$m$$?

Accepted Answer

$$K = \frac{p^2$}{2m}$$

Question 15

Which of the following is a central force in the context of the simple pendulum?

Accepted Answer

Tension

Question 16

If $$\vec{u}$$ and $$\vec{v}$$ are vectors, what is the magnitude of their cross product $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

$$|\vec{u}| |\vec{v}| \sin 	heta$$

Question 17

Which equation represents Newton's second law in terms of momentum?

Accepted Answer

$$\frac{d \vec{p}}{dt} = \sum_{n=1}^N \vec{F}_n$$

Question 18

In the context of angular momentum, what does the vector $$\vec{H}_O$$ physically represent?

Accepted Answer

The rotation of point $$P$$ relative to point $$O$$

Angular Momentum and the Cross Product: Principles, Examples, and Applications

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