Question
Newton's Second Law of Motion in vector form states that $\mathbf{F}=\frac{d \mathbf{p}}{d t},$ where $\mathbf{F}$ is the force acting on an object of mass $m$ and $\mathbf{p}=m \mathbf{r}^{\prime}(t)$ is the object's momentum. The analogs of force and momentum for rotational motion are the torque $\tau=r \times F$ and angular momentum$$\mathbf{J}=\mathbf{r}(t) \times \mathbf{p}(t)$$Use the Second Law to prove that $\tau=\frac{d \mathbf{J}}{d t}$
Step 1
Step 1: We start with the given equation for angular momentum, $\mathbf{J}=\mathbf{r}(t) \times \mathbf{p}(t)$, and take its derivative with respect to time: $$ \frac{d \mathbf{J}}{d t}=\frac{d}{dt}(\mathbf{r}(t) \times \mathbf{p}(t)) $$ Show more…
Show all steps
Your feedback will help us improve your experience
Subham Jyoti Mishra and 57 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
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.
Vector Calculus
Motion on a Curve
If a particle with mass $ m $ moves with position vector $ r(t) $, then its angular momentum is defined as $ L(t) = m r(t) \times v(t) $ and its torque as $ \tau (t) = m r(t) \times a(t) $. Show that $ L^\prime (t) = \tau (t) $. Deduce that if $ \tau (t) = 0 $ for all $ t $, then $ L(t) $ is constant. (This is the law of conservation of angular momentum.)
Vector Functions
Motion in Space: Velocity and Acceleration
Derive the rotational form of Newton's second law as follows. Consider a rigid object that consists of a large number $N$ of particles. Let $F_{i}, m_{i},$ and $r_{i}$ represent the tangential component of the net force acting on the ith particle, the mass of that particle, and the particle's distance from the axis of rotation, respectively. (a) Use Newton's second law to find $a_{i}$, the particle's tangential acceleration. (b) Find the torque acting on this particle. (c) Replace $a_{i}$ with an equivalent expression in terms of the angular acceleration $\alpha$ (d) Sum the torques due to all the particles and show that $$\sum_{i=1}^{N} \tau_{i}=I \alpha$$
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD