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

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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

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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

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Work Energy and Power

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