Chapter 6: Q9P (page 289)

The angular momentum of a particle m is defined by $L = mr \times (\frac{dr}{dt})$ (see end of Section 3). Show that $\frac{dL}{dt} = mr \times \frac{d^{2} r}{{dt}^{2}}$

Short Answer

Expert verified

The proof is given as follows.

\frac{dL}{dt} = mr \times \frac{d^{2} r}{{dt}^{2}}

Step by step solution

Given equation

The angular momentum of a particle m is defined by $L = mr \times (\frac{dr}{dt})$ .

Definition of Angular Momentum

The angular momentum L of m about point O is defined by the equation $L = r \times (mv) = mr \times v$

Proof of the statement

To prove the given condition, Consider,

$L = mr \times (\frac{d^{2} r}{{dt}^{2}})$ ,then:

$\frac{dL}{dt} = m \frac{dr}{dt} \times \frac{dr}{dt} + mr \times \frac{d^{2} r}{{dt}^{2}}$

But since the cross product of a vector with itself results in the zero vector, then:

$\frac{dL}{dt} = mr \frac{d^{2} r}{{dt}^{2}}$

Hence, the result is $\frac{dL}{dt} = m \frac{dr}{dt} \times \frac{dr}{dt} + mr \times \frac{d^{2} r}{{dt}^{2}}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

The angular momentum of a particle m is defined by $L = mr \times (\frac{dr}{dt})$ (see end of Section 3). Show that $\frac{dL}{dt} = mr \times \frac{d^{2} r}{{dt}^{2}}$

Short Answer

Step by step solution

Given equation

Definition of Angular Momentum

Proof of the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Astrophysics

Translational Dynamics

Solid State Physics

Measurements

Medical Physics

Study anywhere. Anytime. Across all devices.

The angular momentum of a particle m is defined by L=mr×(drdt)(see end of Section 3). Show thatdLdt=mr×d2rdt2

Short Answer

Step by step solution

Given equation

Definition of Angular Momentum

Proof of the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Astrophysics

Translational Dynamics

Solid State Physics

Measurements

Medical Physics

Study anywhere. Anytime. Across all devices.

The angular momentum of a particle m is defined by $L = mr \times (\frac{dr}{dt})$ (see end of Section 3). Show that $\frac{dL}{dt} = mr \times \frac{d^{2} r}{{dt}^{2}}$