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Q3.4-20E

Expert-verifiedFound in: Page 90

Book edition
9th

Author(s)
R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages
616 pages

ISBN
9780321977069

**An object at rest on an inclined plane will not slide until the component of the gravitational force down the incline is sufficient to overcome the force due to static friction. Static friction is governed by an experimental law somewhat like that of kinetic friction (Problem 18); it has a magnitude of at most **

Therefore, the object starts sliding down when ${\mathbf{tan\alpha}}_{\mathbf{o}}\mathbf{>}\mathbf{\mu}$.

Here C is the center of mass of an object. As the object starting sliding down

$\left|{\mathbf{mgsin\alpha}}_{\mathbf{o}}\right|\mathbf{>}\left|\mathbf{\mu N}\right|$

However, $\left|\mathbf{\mu N}\right|\mathbf{=}\left|{\mathbf{\mu mgcos\alpha}}_{\mathbf{o}}\right|$

Hence the object starts sliding down when $\left|{\mathbf{mgsin\alpha}}_{\mathbf{o}}\right|\mathbf{>}\left|{\mathbf{\mu mgcos\alpha}}_{\mathbf{o}}\right|$

$\left|{\mathbf{sin\alpha}}_{\mathbf{o}}\right|\mathbf{>}\left|{\mathbf{\mu cos\alpha}}_{\mathbf{o}}\right|$

Assume that the inclined plain is not vertical. Hence, $\mathrm{cos}{\alpha}_{o}\ne 0$ . Then

$\left|\frac{{\mathbf{sin\alpha}}_{\mathbf{o}}}{{\mathbf{cos\alpha}}_{\mathbf{o}}}\right|\mathbf{=}\left|{\mathbf{tan\alpha}}_{\mathbf{o}}\right|$

Both $\mathbf{\mu}$and ${\mathbf{tan\alpha}}_{\mathbf{o}}$are positive,

so $\left|{\mathbf{tan\alpha}}_{\mathbf{o}}\right|\mathbf{=}{\mathbf{tan\alpha}}_{\mathbf{o}}$and $\left|\mathbf{\mu}\right|\mathbf{=}\mathbf{\mu}$

**Therefore, the object starts sliding down when role="math" localid="1664209459647" ${{\mathbf{tan\alpha}}}_{{\mathbf{o}}}{\mathbf{>}}{\mathbf{\mu}}$.**

** **

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