Suppose m is the mass of a moving particle. Newton's second...

Suppose $m$ is the mass of a moving particle. Newton's second law of motion can be written in vector form as $$ \mathbf{F}=m \mathbf{a}=\frac{d}{d t}(m \mathbf{v})=\frac{d \mathbf{p}}{d t} $$ where $\mathbf{p}=m \mathbf{v}$ is called linear momentum. The angular momentum of the particle with respect to the origin is defined to be $\mathbf{L}=\mathbf{r} \times \mathbf{p}$, where $\mathbf{r}$ is its position vector. If the torque of the particle about the origin is $\boldsymbol{\tau}=\mathbf{r} \times \mathbf{F}=\mathbf{r} \times d \mathbf{p} / d t$ show that $\tau$ is the time rate of change of angular momentum.

Suppose $m$ is the mass of a moving particle. Newton's second law of motion can be written in vector form as
$$
\mathbf{F}=m \mathbf{a}=\frac{d}{d t}(m \mathbf{v})=\frac{d \mathbf{p}}{d t}
$$
where $\mathbf{p}=m \mathbf{v}$ is called linear momentum. The angular momentum of the particle with respect to the origin is defined to be $\mathbf{L}=\mathbf{r} \times \mathbf{p}$, where $\mathbf{r}$ is its position vector. If the torque of the particle about the origin is $\boldsymbol{\tau}=\mathbf{r} \times \mathbf{F}=\mathbf{r} \times d \mathbf{p} / d t$
show that $\tau$ is the time rate of change of angular momentum.

Key Concepts

Newton's Second Law

This fundamental law connects the net force acting on an object to the time rate of change of its momentum, establishing that F = dp/dt. It is essential for understanding how forces influence the motion of objects by dictating that a net force results in an acceleration proportional to the mass of the object.

Angular Momentum

Angular momentum is defined as the cross product of the position vector and the linear momentum of an object (L = r × p). It is a measure of the rotational motion of an object and plays a similar role in rotational dynamics as linear momentum does in linear motion.

Torque

Torque is the rotational equivalent of force and is defined as the cross product of the position vector and the force vector (? = r × F). It quantifies the tendency of a force to cause an object to rotate about a point or axis.

Differentiation of Cross Products

When differentiating the angular momentum, which is a cross product of two time-dependent vectors, the product rule for derivatives is applied. This rule shows that the derivative of a cross product equals the cross product of one vector with the derivative of the other plus the cross product of the derivative of the first vector with the other. In the context of angular momentum, applying this rule reveals that one term vanishes (because the velocity is parallel to momentum), leading to the conclusion that the time rate of change of angular momentum equals the torque.

Transcript

Please give Ace some feedback