Q37P 聽In classical mechanics, Newton... [FREE SOLUTION]

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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 12: Q37P (page 549)

In classical mechanics, Newton鈥檚 law can be written in the more familiar form $F = ma$ . The relativistic equation, $F = \frac{dp}{dt}$ , cannot be so simply expressed. Show, rather, that
$F = \frac{m}{\sqrt{1 - u^{2} / c^{2}}} [a + \frac{u (u 鈥� a)}{c^{2} 鈥� u^{2}}]$
where $a = \frac{du}{dt}$ is the ordinary acceleration.

Short Answer

Expert verified

It is proved that $F = \frac{m}{\sqrt{1 \frac{u^{2}}{c^{2}}}} \{a + \frac{u (u 鈥� a)}{c^{2} - u^{2}}\}$

Step by step solution

Expression for the realistic momentum:

Write the expression for the realistic momentum is given by,

$p = \frac{mu}{\sqrt{1 + \frac{u^{2}}{c^{2}}}}$ 鈥︹€� (1)

Here, m is the mass, u is the ordinary velocity, and c is the speed of light.

Show that :

From the given problem, it is given that:

$F = \frac{dp}{dt} Also, a = \frac{du}{dt}$ .......(2)

Substitute the value of equation (1) in equation (2).

$F = \frac{d}{dt} [\frac{mu}{\sqrt{1 - \frac{u^{2}}{c^{2}}}}] F = m [\frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} \frac{du}{dt} + (- \frac{1}{2}) \frac{- \frac{1}{c^{2}} (2 u) \frac{du}{dt}}{(1 - \frac{u^{2}}{c^{2}})}]$

On further solving,

$F = [\begin{matrix} \end{matrix} \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} a + \frac{\frac{1}{c^{2}} u (u - a)}{(1 - \frac{u^{2}}{c^{2}})}] F = [\begin{matrix} \end{matrix} \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} a + \frac{\frac{1}{c^{2}} u (u 鈥� a)}{{(\frac{c^{2} - u^{2}}{c^{2}})}^{\frac{3}{2}}}] F = \frac{m}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} [a + \frac{u (u - a)}{(\frac{c^{2} - u^{2}}{c^{2}})}]$

Therefore, it is proved that $F = \frac{m}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} [a + \frac{u (u - a)}{(\frac{c^{2} - u^{2}}{c^{2}})}]$

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Short Answer

Step by step solution

Expression for the realistic momentum:

Show that :

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In classical mechanics, Newton鈥檚 law can be written in the more familiar form F=ma. The relativistic equation, F=dpdt, cannot be so simply expressed. Show, rather, thatF=m1-u2/c2[a+uu鈥�ac2鈥�u2]where a=dudt is the ordinary acceleration.

Short Answer

Step by step solution

Expression for the realistic momentum:

Show that :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Waves Physics

Electric Charge Field and Potential

Torque and Rotational Motion

Engineering Physics

Translational Dynamics

Fluids

Study anywhere. Anytime. Across all devices.